Put chocolates and butter in a medium metal bowl and set bowl over a pan filled with 1 in. of simmering water. Cook, stirring occasionally, until melted, then remove from heat and let cool slightly. Whisk in eggs and sugar, mixing until combined. Then whisk in flour, baking powder, and salt. Chill dough, covered, until firm, about 2 hours.
Let dough sit at room temperature 15 minutes. Meanwhile, preheat oven to 350° and line 2 baking sheets with parchment paper. Scoop 1 tbsp. portions of dough, rolling each into a ball, and put onto sheets 1 in. apart. Using the palm of your hand, press dough balls into rounds 1/4 in. thick.
Bake cookies until they no longer look wet on top, about 8 minutes. Let cool on baking sheets.
In a small bowl, mix powdered sugar, milk, and peppermint extract until smooth. Spread 1 heaping tsp. peppermint icing onto the flat side of 1 cookie. Top with flat side of a second cookie to form a sandwich, pressing together to squeeze filling to the edge. Roll edge of cookie in candy canes. Repeat with remaining cookies.
Make ahead: Batter up to 1 day; baked cookies up to 2 days, airtight.
Note: Nutritional analysis is per sandwich cookie.