Place a rack in middle of oven; preheat to 350°. Whisk 1 cup (125 g) all-purpose flour, ½ tsp. baking powder, and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt in a small bowl to combine. Using an electric mixer on medium-high speed, beat ½ cup (100 g) granulated sugar and 6 Tbsp. unsalted butter, room temperature, in a large bowl until light and fluffy, about 2 minutes. Add 1 large egg yolk; beat to combine. Add ⅓ cup sour cream and ¼ tsp. peppermint extract; beat again to combine. Add dry ingredients, reduce mixer speed to low, and beat until well combined. Using a rubber spatula, mix until no dry streaks of flour remain.

Using a #40 cookie scoop (about 2 Tbsp.), portion out dough, dividing between 2 baking sheets lined with silicone baking mats (for the smoothest bottoms) or parchment paper, spacing about 2" apart (up to 8 per baking sheet). Working with 1 baking sheet at a time, bake cookies until puffed, matte on top, and golden underneath, 14–16 minutes (check after 12 minutes if using parchment). Let cool on baking sheets 5 minutes. Carefully transfer to a wire rack; let cool completely. Do Ahead: Cookies can be baked 1 day ahead. Store airtight at room temperature.

Place 2 full-size candy canes in a resealable plastic bag. Seal and lightly crush candy into small pieces with a rolling pin or skillet (try not to overdo it or you’ll end up with powder). Set aside.

Stir 3 cups (330 g) powdered sugar, ¼ tsp. peppermint extract, 2 Tbsp. light corn syrup, and 2 Tbsp. water in a small bowl until smooth (glaze should be very thick but fluid; add more water 1 tsp. at a time up to 1 Tbsp. to loosen or drizzle in more corn syrup as needed to thicken). Pour half of glaze into another small bowl. Add ¼ cup Dutch-process cocoa powder, remaining 1 Tbsp. light corn syrup, and 1 Tbsp. water; stir, adding more water 1 tsp. at a time if needed, until cocoa powder is dissolved and glaze is smooth and thick.

Turn cookies over. Using a small offset spatula, spread one half of each cookie with about ½ tsp. white glaze. Repeat with chocolate glaze on empty half of each cookie.

Immediately, lightly sprinkle one side of each cookie with reserved crushed candy canes (alternate between white and chocolate glaze if desired). Let cookies sit at room temperature until glaze is no longer wet to the touch, 30–45 minutes. Do Ahead: Cookies can be made 12 hours ahead. Store airtight at room temperature.