Preheat oven to 350 degrees F (175 degrees C). Line a 12-cup muffin tin with paper liners.
Beat white sugar, brown sugar, and butter together in the bowl of a stand mixer until thoroughly blended and creamy. Add egg; beat well. Mix in 1 teaspoon vanilla extract.
Combine flour, 2/3 cup cocoa powder, baking powder, coffee, salt, baking soda, and cinnamon in a second bowl. Blend with a whisk until fluffy and no clumps appear. Alternate adding the flour mixture and 1 cup milk into the butter mixture, mixing until just combined. Fill muffin cups 2/3-full of batter.
Bake in the preheated oven until a toothpick inserted into the center of a cupcake comes out clean, 15 to 17 minutes. Move to a wire rack to cool completely, at least 25 minutes.
Beat butter in the bowl of the stand mixer; add confectioners' sugar, 1 cup at a time, beating until creamy. Add 2 tablespoons milk; mix until frosting is creamy. Mix in 1/2 cup cocoa powder and 1 teaspoon vanilla extract.
Fit a piping bag with a coupler and tip. Drop piping bag into a tall glass and fold edges down around the glass. Spoon frosting into the bag. Gather edges and twist together. Pipe frosting onto the cupcakes.
