Preheat oven to 350°.
To prepare cupcakes, beat first 3 ingredients at medium speed of a mixer until well-blended (about 3 minutes). Add eggs, 1 at a time, beating well after each addition. Lightly spoon flour into a dry measuring cup, and level with a knife. Combine flour, cocoa, baking soda, and salt, stirring well with a whisk. Add flour mixture to sugar mixture alternately with 1/2 cup milk, beginning and ending with flour mixture; mix after each addition.
Spoon the batter into 12 muffin cups lined with paper liners. Bake at 350° for 22 minutes or until cupcakes spring back easily when touched lightly in the center. Cool in pan 10 minutes on a wire rack, and remove from pan. Cool completely on wire rack.
To prepare frosting, melt 1 tablespoon butter in a small, heavy saucepan over low heat. Add 1/4 cup granulated sugar, 3 tablespoons milk, and chopped chocolate, and cook 3 minutes, stirring constantly. Remove from heat, and cool. Stir in the powdered sugar and 1 teaspoon vanilla. Spread frosting over cupcakes.