Here is the full list of all 15 Idle Cooking Emperor achievements.
Evolve one Fairy two times
Winn 1000 cooking battles
Beat at least one Pupil's Pupil Challenge
Have 100 pupils
Beat at least one Challenger's Challenge
Kill a judge
Have at least 3 fairies
Beat the opponent in area 30
Win your first Cooking Battle
Have at least 1000 Research Levels
Have 1 million fame
Play for 30 days
Have 100 million dollars
Have 25,000 total stats
Have 10,000 experience