Here is the full list of all 15 Idle Cooking Emperor achievements.
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
Evolve one Fairy two times
Kill a judge
Beat the opponent in area 30
Have at least 3 fairies
Have 100 pupils
Winn 1000 cooking battles
Beat at least one Pupil's Pupil Challenge
Beat at least one Challenger's Challenge