These cakey cookies can dry out if over-baked, so keep a close eye while they are in the oven. Make sure to buy peppermint extract and not mint (which has more of a toothpaste quality to it) and gradually add to taste in the filling. The minty flavor will grow as it sits, so if you are making these ahead of time, you may want to add the lesser amount.
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.