Preheat oven to 350 degrees. Line a large baking sheet with parchment.
In a large bowl, cream the butter and sugars for 2-3 minutes until smooth and fluffy. Beat in the egg, then the egg yolk, until each are fully incorporated. Whip in the vanilla and peppermint extracts.
In a separate small bowl, whisk together the flour, baking soda, and salt. Add in two batches to the wet ingredients, mixing each until just combined. Fold in the dark and white chocolate chunks, and the peppermint bark.
Scoop 1 1/2-2 tablespoon sized balls of dough onto the baking sheet, leaving 2-3 inches in between each. Bake for 8 minutes, then tap the baking sheet on the counter to flatten the tops. Sprinkle with additional white chocolate and crushed candy cane bits. Bake an additional 3-4 minutes until the edges are set but the centers are still doughy.
Cool cookies on the baking sheet. Serve warm, or store in an airtight container for up to 3 days.