NHL 21

Much easier method in my opinion. Two controllers required:

1. In the main menu, go to NHL Threes
2. Pick Threes Now (so you can play offline)
3. With the controller that will get the achievement, pick a side and any team. Hit the left stick click to lock your position to goalie.
4. With the other controller, pick the other side and any team.
5. Start the match.
6. During the match, don’t touch the controller that will get the achievement. Instead, use the other controller to keep giving the puck away. And pass the puck away whenever you have the puck, and skate away from the play or from your goalie. Because the team that will get the achievement is position-locked to the goalie, the three skaters will be controlled by CPU. Allow the team that will get the achievement to simply score goals on its own with the CPU skaters.
7. After only a few minutes the team that will get the achievement will have easily scored 5 goals with its CPU skaters. Then the game will automatically end, and the achievement will pop.

You can snag this pretty easily in Franchise mode without having to do much work by jumping into the end of a simulated game.

Sim the first 2 periods of any given game, then slow sim to the final minute of the 3rd. If your team lets in a goal at any time, go ahead and move on to the next game. Intervene with less than a minute left, on 8x sim speed it should leave you with around 30 seconds left. Bump the difficulty down to rookie before you start the game. Make sure to position lock yourself into goalie or this won't work.

Once you're in the game the faceoff will be at center ice. You'll maybe have to make a save or two, as the other team will most likely pull their goalie if the game is close. After the clock runs out, you'll have your shutout and your achievement.

Another solution to this achievement. I played it in Pro-Am in World of Chel. I did one of the Best of 3 games. My team ended up scoring 3 goals and I had to only make maybe 2-4 saves. Achievement popped after game ended.