Preheat the oven to 175 degrees Celsius. Line cupcake liners in a muffin tin or on a baking sheet.
Whisk flour, baking powder, salt and instant coffee together in a small bowl. Set aside. Combine the cooled coffee and milk in a small cup and set aside.
Beat the butter and sugar with an electric mixer until light, pale and fluffy, about 5 minutes. Scrape down the sides of the bowl. Beat in the eggs, one at a time, adding the next one only when the previous one has been fully incorporated. Scrape down the sides of the bowl.
Sift in 1/3 of the flour mixture and beat it in on low speed until some pockets of flour remain. With the mixer speed running on low, add in 1/2 of the coffee-milk mixture in a slow, steady stream. Beat until fully incorporated. Sift in 1/2 of the remaining flour mixture and beat it in on low speed. Add in the rest of the coffee-milk mixture in a slow, steady stream, beating it in on low speed. Sift in the rest of the flour mixture and mix until all the flour has been incorporated.
Spoon the batter into the cupcake liners until 3/4 full. For cupcakes that require 1/4-cup batter (regular-sized cupcakes), bake them for 20 to 25 minutes; for cupcakes that require 1/8-cup batter (mini-sized cupcakes), bake them for 18 to 20 minutes; until the cupcakes are well-risen and a skewer inserted into the centre of the cake comes out clean. Cool the cupcakes completely on a wire rack before frosting.
Frosting
In a medium bowl, combine the butter, cocoa powder, 125 grams (1 cup) icing sugar and the dissolved coffee. Beat with an electric mixer until light and fluffy. Add in more icing sugar to suit your taste. Beat in the coffee liqueur and milk for a softer consistency. Frost as desired.