Given the title of the puzzle and the phrases calm clam and post stop in the flavor text, we can tell that this puzzle has something to do with anagramming.

First, we solve the clues. Incidentally, I’ve shown some serious mercy with how unambiguous most of them are.

From

  • HANS
  • BAKE
  • RAGE
  • ONCE
  • SEAM
  • OPUS
  • MAZE
  • ALUM
  • FILE
  • DISK
  • RACE
  • UTAH
  • FOAL
  • TRIO
  • SUEZ
  • TIDE
  • TUNI
  • DONE
  • LEAF
  • SLOT
  • SALT
  • NEAT

To

  • LAST
  • SKID
  • BEAK
  • GEAR
  • NASH
  • ZEUS
  • MAZE
  • FLEA
  • ANTE
  • DIET
  • ACRE
  • SOUP
  • LOST
  • CONE
  • TORI
  • OLAF
  • MESA
  • UINT
  • UTAH
  • NODE
  • LIFE
  • MAUL

We notice that the words in the To column are anagrams of the words in the From column. Given that the clues of the To column were in alphabetical order, the ones in the From column contain the order that matters. So we rearrange the To column:

From

  • HANS
  • BAKE
  • RAGE
  • ONCE
  • SEAM
  • OPUS
  • MAZE
  • ALUM
  • FILE
  • DISK
  • RACE
  • UTAH
  • FOAL
  • TRIO
  • SUEZ
  • TIDE
  • TUNI
  • DONE
  • LEAF
  • SLOT
  • SALT
  • NEAT

To

  • NASH
  • BEAK
  • GEAR
  • CONE
  • MESA
  • SOUP
  • MAZE
  • MAUL
  • LIFE
  • SKID
  • ACRE
  • UTAH
  • OLAF
  • TORI
  • ZEUS
  • DIET
  • UINT
  • NODE
  • FLEA
  • LOST
  • LAST
  • ANTE

If we get Stuck On Extraction™, we would solve “The Chess Fairy” first, which would give us COMBINE PERMUTATIONS along with the elaboration that allows us to perform the following extraction:

Permutation

  • 3241
  • 1423
  • 3421
  • 3124
  • 4213
  • 4132
  • 1234
  • 4132
  • 3214
  • 3421
  • 2314
  • 1234
  • 2431
  • 1423
  • 4321
  • 3241
  • 2431
  • 3214
  • 4123
  • 2314
  • 3214
  • 3142

Letter

  • P
  • E
  • R
  • M
  • U
  • T
  • A
  • T
  • O
  • R
  • I
  • A
  • L
  • E
  • X
  • P
  • L
  • O
  • S
  • I
  • O
  • N

So the answer is PERMUTATORIAL EXPLOSION.

Notes:

Perhaps I should have chosen a less reasonable extraction method, as one out of three teams managed to figure out the method without solving a different puzzle first. Also, maybe the clues were a little too merciful.