Allow butter and eggs to stand at room temperature for 30 minutes. Meanwhile, preheat oven to 350 degrees F. Grease and flour twenty-eight 2 1/2-inch muffin cups or line with paper bake cups. In a medium bowl, stir together flour, 1/2 cup cocoa powder, espresso powder, baking soda, baking powder, and salt; set aside.
In a large bowl, beat butter with an electric mixer on medium to high speed for 30 seconds. Gradually add sugar, beating on medium speed until combined. Beat for 2 minutes more, scraping side of bowl occasionally. Add eggs, one at a time, beating well after each addition. Beat in liqueur and vanilla. Alternately add flour mixture and chocolate milk, beating on low speed after each addition just until combined. Beat on medium to high speed for 20 seconds more. Spoon batter into the prepared cups, filling each about three-fourths full.
Bake in the preheated oven for 15 to 18 minutes or until tops spring back when lightly touched. Cool in pans on wire racks for 5 minutes. Remove from pans; cool completely on racks.
Increase oven temperature to 450 degrees F. Pipe or spread Marshmallow-Coffee Frosting onto tops of cupcakes. Place frosted cupcakes on a large baking sheet. Bake in the preheated oven about 5 minutes or until frosting is lightly browned. Cool completely on wire racks.