1. Preheat the oven to 350°F. Line two baking sheets with parchment.

  2. To make the dough: In a large mixing bowl, beat together the butter, sugar, baking powder, and salt until fluffy.

  3. Add the vanilla and flour and mix until combined. Scoop the dough into 1" balls weighing 3/8 ounce (10.5g) each.

  4. Place the dough balls onto the prepared baking sheets, leaving 1 1/2" between them. Use the bottom of a glass dipped in sugar to flatten the balls to about 1/4" thick.

  5. Bake the cookies until they're light brown around the edges, 9 to 11 minutes. Remove them from the oven and let them cool on the pan on a rack.

  6. To make the filling: In a small saucepan, melt the butter over low heat until it separates and starts to simmer. Cook until the liquid portion of the butter is clear, not cloudy. There will be foamy-looking solids floating in the butter; that's okay. If they start to brown, turn the heat down.

  7. In a large mixing bowl, combine the shortening, confectioners' sugar, salt, and melted butter. Mix until the filling comes together; add the vanilla and peppermint extracts.

  8. To assemble: Turn half the cookies over. Take a tablespoon of filling, roll it into a ball, and flatten it into a 2" disk. Place on the bottom of one cookie and top with a second, pressing down gently.

  9. Roll the edges of the cookies in the crushed candy canes (or pearl sugar).

  10. Store cookies in an airtight container at room temperature for up to 1 week; freeze for longer storage.