Preheat oven to 325 degrees F (165 degrees C). Grease or line 20 muffin cups with paper liners.
Beat white sugar and 7 tablespoons butter together in a bowl using an electric mixer until smooth and creamy; add eggs, vanilla extract, and baking powder and beat until smooth. Sift flour into creamed butter mixture until batter is smooth. Spoon 2 tablespoons batter into each muffin cup.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, about 15 minutes. Cool cupcakes in the tin for 3 minutes before transferring to a wire rack to cool completely.
Mix confectioners' sugar and 1/2 cup butter together in a bowl until frosting is smooth; spread onto cooled cupcakes.