Beat butter and 1 cup confectioners' sugar in a large bowl until creamy and smooth; stir peppermint extract into butter mixture. Whisk flour and salt in a bowl and stir into butter mixture. Gently fold rolled oats and 1/3 cup crushed peppermint candies into the dough.
Drop dough by teaspoonful onto ungreased baking sheets, leaving 3 inches between cookies; press cookies 1/4 inch thick with your fingertips.
Bake in the preheated oven until cookies have lightly browned edges, 10 to 12 minutes. Let cookies cool on baking sheets for at least 5 minutes before removing to finish cooling on wire racks.
Stir 2 tablespoons confectioners' sugar with 1 tablespoon very finely crushed peppermint candies in a small bowl; transfer to a fine-mesh wire strainer. Dust cooled cookies with peppermint candy mixture.
