Position racks in the upper and lower thirds of the oven and preheat to 375 degrees F. Line 2 baking sheets with parchment paper and coat with cooking spray. Whisk the flour, baking powder, baking soda and 1/4 teaspoon salt in a medium bowl. Beat 6 tablespoons butter and the granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the egg and beat until combined. Reduce the mixer speed to low; beat in the flour mixture in two batches, alternating with the milk and vanilla, until just combined.
Arrange mounds of batter (about 1 heaping tablespoonful each) about 2 inches apart on the prepared baking sheets; shape into rounds with damp fingers. Bake, switching the pans halfway through, until the tops of the cookies spring back when pressed, 9 to 12 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Meanwhile, beat the remaining 6 tablespoons butter, the peppermint extract and the remaining pinch of salt in a large bowl with a mixer on medium speed until creamy, about 2 minutes. Gradually beat in the confectioners' sugar until smooth. Spread 1 1/2 tablespoons filling on the flat side of half of the cookies; sandwich with the remaining cookies. Roll the edges in the crushed candy canes.