Crusader Kings III

There is no 5 minute solution for this, because you'll need advanced technology to build the necessary holding upgrades to push the income from that one single holding up to 60 gold/month. You'll probably need the late medieval technologies "machicolations" (for building a level 4 castle) and cranes (for building the late medieval upgrades for economy buildings). I unlocked the achievement in the late game during a long term playthrough in which I also aimed to unlock "beacon of progress" and to reunite the Roman Empire, so I wasn't in a hurry ;-).

EDIT: I have since experimented a bit, it is in fact possible to speedrun this using only high medieval technologies, but you'll need one of the very best mines and stack every tax bonus possible, see chapter 4 for details.


Here is what you need to do

1) Finding a holding with a good mine

Mines are special buildings (similar to holy sites) that can only be built in some few select counties that have a special building slot for them. You can find them by using the economy map (one of the lesser used maps reached via the + map button). When you zoom in, you can see every existing (yellow) or potential (blue) special building and hover over the building symbol to view either the stats of their current level (if they are already built) or their first level, if they aren't build yet. Most mines aren't built yet at the start of the game.

Be careful, NOT ALL MINES HAVE THE SAME STATS!
Mines provide various individual boni. In terms of their income bonus (which is what we are looking for above anything else to get that 60 income), there are basically three kinds of mines, the worst (usually salt or copper mines) start with an income bonus of 2, the mediocre start at 3 and the best start at 5. The latter category also all have a precentage bonus on the holding tax income that starts at 20% and upgrades up to 35% at level 4. Each mine can be upgraded 3 times, the upgrades require the castle upgrade of the same level. The income bonus develops as follows:

bad mines: 2 - 3.5 - 5 - 6.5
mediocre mines: 3 - 5 - 7 - 9
best mines: 5 - 8 - 11 - 14

If you plan to play a long late middle ages campaign anyway, you don't necessarily have to pick one of the best mines, a mediocre one will be sufficient with fully upgraded economy buildings, but don't pick one of the worst category, getting those to 60 income will be very difficult, if not outright impossible (much of the work is done by % multipliers, so having only 6.5 base instead of 9 or 14 does make quite a difference). In my first playthrough, I accidently picked a mediocre one (Trepca mines in Serbia) and I managed to push its income up to 85 in the end (with fully upgraded buildings, about 50 development and a godlike steward skill)

The mines in the best category are: Rammelsberg (central Germany), Srebrenica (Croatia), Kollur mine and Ratnapura mine (both Decan Empire in India) and the 3 gold mines of Mali (West Africa).

Once you seized your mine holding, start developing and upgrading it. You'll eventually have to move your capital there (by clicking on the symbol with a yellow crown and arrow in the county building screen while the holding is part of your personal domain), but you don't have to do that right away if the place is inconvenient for your capital or you have other valuable buildings like holy sites or duchy buildings that you want to keep in your domain at all cost. Even if you lose the holding to a vassal during the game, you can still keep building upgrades there and get it back once you get near the 60 gold income threshold.

2) Getting the holding to 60 gold income

If the mine holding isn't already the capital of the county, first move the county capital there, so you can build those very lucrative buildings there which can only be built in the county capital (like e.g. windmills and watermills).

Fill every building slot with building upgrades that provide flat tax income and/or a tax income % multiplier. Don't waste any building slots on buildings that just provide levies or troop boni. Upgrade them as far as your technology level and your purse allows.

Build the tax collectors duchy building in the county that is the duchy capital of your mine holding, which will give a bonus on taxes for every county within the duchy (you need to also personally hold the duchy capital and the duchy title for that bonus to affect the mine holding)

Push the development of the county by having your steward do county development there. Some of your building upgrades will also help with that. Each point of county development will provide + 0.5 % to tax income.

Push your ruler's stewardship skill as high as possible, since that skill also provides a bonus to your tax income.

Try to get more county multipliers and personal multipliers that provide a bonus on the domain tax income. Boni that just say bonus to "income" like e.g. the 10% from the wealth lifestyle or from the "greedy" character trait won't help, because they are only multiplied with your total income, not with the tax income from individual fiefdoms, but boni that say they increase (domain) tax income will help. Some artefacts and court artefacts have boni on tax income. The finisher trait "avaricious" from the wealth lifestyle skill tree provides 15 % bonus on tax income. The perk "absolute control" (second to last perk of the overseer skill tree) will also give you a bonus on domain taxes and levies in any county where the control level is at 100%.

After moving your capital to the mining holding, you can do a bit of traveling until you encounter the fairly common travel event "other people's pastures", in which you pass through land with particularly prosperous pastures. Pick "Maybe I could apply some of this at home" and pass the stewardship skill check to get the county modifier "improved pastures" in your capital, which gives a temporary + 20% on tax income.

For the last push, you can switch your steward's councillor task to collecting taxes, which will also provide a bonus to your domain taxes depending on the stewards skill. The starting perk "tax (wo)man" from the architect skill tree boosts the effect of this councillor task by 25%.

3) Stationing 500+ armored cavallery in the holding

This is the easiest part. Armoured riders are unlocked by the early medival innovation "arched saddle". Cavallery is only 50 men per regiment size, thus you need a cavallery regiment of at least size 10. Depending on your culture, you might need to either unlock the high medieval "men at arms" technology and/or get an acclaimed knight with the lancer attribute to get your regiment size cap to 10+, but by the time you are at a high enough technology level to get the 60 gold income, doing this should be no problem. Station that regiment in the holding.

4) Speedrunning this with only high medieval technology

You need to pick one of the very best mines if you want to do this without late medieval technology. A good place to do this quickly is the kingdom of Lanka (Island of Sri Lanka, South of India). Start as the ruler who controls the South of the Island in 1066, you will start involved in a historical Rebellion against the Tamil with a lot of special troops that make the war easy to win. After winning the war, you'll be in control of the whole Island, with them mine in the province of Malaya already build and a holy site in Pihiti that doubles as an already built university. Unfortunately, Malaya is in the hand of a vassal, so either fabricate a claim or take those tyranny points and steal it from him, which will immediately put your gold income into the double digits.
With those two special buildings in your domain and overall highly developed land in your culture and only one direct neighbour, Lanka is an ideal place to play tall. You could stay just on the island, but I'd recommend to fabricate a few claims and take another chunk out of the neighbouring Chola Kingdom, both to prevent them from becoming too strong and to get some even higher developed counties of the Tamil culture. Creating a hybrid culture with them will save you some research time and raise the average development of your culture even higher.
When educating your heirs, go first for learning, then stewardship when you pushed the research far enough. Have scholarship lifestyle most of the time for the development bonus. As a minimum you'll need the high medieval technologies windmills, guilds and hoardings, men-at arms isn't strictly necessary if you have a lancer accolade to get the 500 heavy cavallery regiment. Always have your steward push development in the mine county until you are ready to break the 60 gold income threshold by switching him to tax collecting.

I managed to get to 60 income around the year 1150 with about 40 development, the avaricious and taxmen perks, very good stewardship and the following buildings built up to their maximum level at high medieval technology (5 or 6): Windmills, hunting grounds, cattle pastures, hillside farms, hospices and blacksmiths and of course the castle and the mine at level 3. I also had a level 2 tax collector in the neighboring duchy capital and the "improved pastures" modifier from the travel event.