In a large bowl, beat butter, sugar, egg and vanilla. Combine flour, cocoa powder, baking powder and salt; gradually mix into butter mixture. If dough is very soft, refrigerate for 30 minutes. Divide dough in half. Shape into two 6x2-inch rolls; wrap in plastic wrap. Refrigerate or freeze until firm enough to slice, 1-2 hours, or up to two weeks.
Cut one roll into 24, 1/4-inch slices. Place 12 slices on a baking sheet. Top each slice with a peppermint patty. Top with another slice of dough and press edges of dough together to completely cover patty.
Cut remaining roll into 12, 1/2-inch slices. (These will be the bottoms of the sandwiches).
Bake in preheated 325°F for 10-12 minutes until tops feel set when touched. Cool 5 minutes on pan. Remove to wire rack to cool completely.
Combine remaining 8 peppermint patties and butter in a small microwave-safe bowl. Heat on medium power 1-2 minutes; stir until melted. Use a small spoon or tip of a fork to drizzle chocolate over the mint-filled cookies. Let cool until icing is hardened.
Meanwhile, make 12 equal size scoops of ice cream and place on a metal pan. Use a pancake turner to flatten each scoop to about 5/8-inch thickness. Return to freezer until firm.
To assemble sandwiches, slide spatula under a flattened scoop of ice cream and place on a plain cookie. Top with an iced cookie. Put into a covered container and freeze. Repeat with remaining cookies and ice cream. Keep on hand for a ready-made dessert or after-school snack.
Fun Tip: Scoop and flatten 4 or 6 scoops of ice cream at a time and return to freezer to prevent melting. Use another tray and repeat for remaining scoops.