1. Working in two batches, place candy canes or peppermint candy in the bowl of a food processor fitted with the metal blade, and pulse several times until finely chopped. (Alternatively, place on a large cutting board, cover with plastic wrap, and crush with a kitchen mallet.) Using a sieve, sift out and discard powdery residue. Measure 1 cup chopped candy; reserve the rest for rolling.

2. In a small saucepan, bring cream, butter, and corn syrup to a full boil over medium heat. Turn off heat. Add 1 pound chocolate; gently swirl pan to cover chocolate with cream, but do not stir. Let stand, undisturbed, 5 minutes. Stir in peppermint extract.

3. Slowly whisk until combined. Transfer mixture to a large bowl; refrigerate, stirring every 15 minutes.

4. After 45 minutes, mixture will thicken quickly, so stir every 3 to 5 minutes until thick enough to scoop, 10 to 20 minutes more. Stir in chopped candy. Using two spoons or a small ice-cream scoop, form into 1-inch balls; transfer to a parchment-lined baking sheet. Return to refrigerator for 10 to 20 minutes, stirring every 3 to 5 minutes.

5. Remove from refrigerator; roll in palms to form a ball, and press gently with fingers to create irregular shapes. Chill until ready to dip, up to 1 week. Place remaining 12 ounces chocolate in a heatproof bowl over a pan of simmering water; stir occasionally until chocolate melts. Remove from heat; cool slightly.

6. Place reserved chopped candy in a small bowl. Remove centers from refrigerator. Using one hand, dip one center into melted chocolate, then roll around in your hand to coat evenly, letting excess drip back into pan. Place truffle in chopped candy. With your clean hand, cover truffle with the candy. Let sit in bowl 20 seconds.

7. Lift out, and set on a parchment-lined baking sheet. Repeat with remaining truffles. If your kitchen is warm, refrigerate 5 minutes to set. You can store truffles in airtight containers up to 1 week at a cool room temperature.