Combine the flour, sugar, salt, baking powder and 2 sticks of the butter in a large bowl. Beat with an electric mixer until the mixture resembles coarse sand.
Add the milk, vanilla and egg and beat again until the dough just comes together.
Transfer the dough to a lightly floured surface and gently push together until a solid ball of dough forms. Transfer the dough to a sheet of parchment paper and form into a log 2 1/2 to 3 inches wide. Roll the dough tightly in the parchment paper and twist the ends closed. Refrigerate until firm and thoroughly chilled, at least 1 hour.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper. Slice the chilled dough into disks 1/4-inch thick. Place the slices on the parchment-lined baking sheets with 1 1/2 to 2 inches of space between each cookie, about 8 cookies per sheet. Keep the rest of the roll in the refrigerator as the first batch bakes. Bake until lightly browned around the edges and slightly soft in the center, about 12 minutes. Place the pans on a rack to cool. Repeat with the remaining dough.
While the cookies cool, combine the chocolate and the remaining 2 tablespoons butter in a heatproof bowl set over a pan of gently simmering water. Let sit until the chocolate looks almost melted. Stir until smooth.
Hold the cookies by the edges and carefully dip the surface of each cookie into the melted chocolate about halfway. Place the cookies on a rack chocolate-side up. Sprinkle each cookie with pieces of peppermint or pretzel. Cool completely (either at room temperature or in the refrigerator) to set before serving.