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1. In a large bowl, beat the 3/4 cup butter with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar and baking powder; beat until combined. Beat in egg. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour.
2. Divide dough in half. Place each dough half between two sheets of waxed paper; roll each to a 12x10-inch rectangle. Chill dough rectangles for 30 minutes.
3. Preheat oven to 350 degrees F. Remove top sheets of waxed paper. Using a fluted pastry wheel or pizza cutter, cut each dough rectangle lengthwise into 10 strips and crosswise into 6 strips to make sixty 2x1-inch rectangles per dough half (120 rectangles total). Transfer dough rectangles to an ungreased cookie sheet, leaving 1 inch between rectangles. (If necessary, chill dough about 15 minutes more before transferring.) Bake in preheated oven for 7 to 9 minutes or until edges are very lightly browned. Transfer cookies to a wire rack; let cool.For frosting::
4. In a medium bowl, beat the 1/4 cup butter on medium to high speed for 30 seconds. Add 1 cup of the powdered sugar, 1 tablespoon of the milk, and the peppermint extract; beat until smooth. Beat in the remaining 1 cup powdered sugar and enough of the remaining milk to make a frosting of spreading consistency. If desired, use food coloring to tint frosting pink.
5. To assemble, pipe or spread a little of the frosting on the bottom of one cookie; top with a second cookie, flat side down, and press lightly together. Pipe or spread with more of the frosting; top with a third cookie, flat side down. Repeat with the remaining cookies and the remaining frosting. Pipe dots of melted chocolate around edges of each assembled sandwich cookie or drizzle the melted chocolate over cookies.* Chill about 30 minutes or until chocolate is set. Makes about 36 sandwich cookies.