Here is the full list of all 25 Battle Chef Brigade achievements.
Achieve 10 Daily Cook-Off victories
Reach a combo multiplier of 10
Beat the game without losing a match
Complete Story Mode
Cure Ardea
Complete all of Pontida's Jobs
Complete all of Thorn's Hunts
Complete all of Belchior's Puzzles
Use Thrash's Heavy Attack to smash a monster into 3 others
Consume 10 ingredients in one match
Join the Brigade as Mina
Stock your pantry with 30 ingredients in one match
Deflect 10 projectiles with a heavy attack
Achieve a total round score of 1000 points or more
Win a round with over 2 minutes remaining on the match clock
Make a dish worth 450 points or more
Buy 15 items
Obtain a sample of Ambrosia
Obtain extra points from a Judge for using any Expertise Tome 5 times
Win a duel without using the theme ingredient
Kill 3 enemies without touching the ground
Jump on the bed 10 times
Win the practice match against Thrash
Reach Brigade Rank 12 in Daily Cook-Off
Complete Story Mode on Hard