Here is the full list of all 10 Two Move Chess achievements.
Clear all white stages
Clear all black stages
Lose a queen, and then win
Promote two pawns to bishop pair
Play f6, and then win with black
Win Connect 5 diagonally with black
Underpromote to a knight, and then win with white
Lose a queen for questionable compensation
Collect the cake which appears with low probability
Play castles check with black