This achievement should be tackled after completing all of the 5x5, 10x10 and 15x15 main puzzles since those are also tied to achievements. Once those are all completed you'll have done 366 puzzles already.
After this the only way to continue progressing the counter is by doing workshop puzzles. To speed up this achievement, only do 5x5 puzzles since you can solve them in 5-20 seconds usually.