Second Extinction

Mission: Vestige (fewest objectives to complete before extraction compared to other missions)
Character choice: Rosie (her ult is a full heal. As long as you're dealing damage you'll replenish your ult quickly so use that ability liberally)
Loadout: Minigun, Ammo Kit, Equipment supply, Health stim (if you run out of ammo use a kit, if you run out of kits then drop your Equipment supply, repeat)
Tips: This mission consists of finding some wreckage, clicking on a body nearby, then dropping a scanner and following the laser pointer to a big cave. Then drop carriers to pick up 4 different T-rex eggs. Momma Rex will show up but you don't have to defeat her to complete the mission. Just grab each egg (1 at a time) and return, repeat. Then just get to an extraction point. If you're struggling at extraction just run in circles around the platform until the ship arrives, the AI will usually have trouble pinning you down.

Edit: Unlocking this in the Tutorial was patched on May 24th, 2021.
This can currently be done by just completing the Tutorial solo.
Just select Gateway Cygnus - Tutorial & Training from the main menu.