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