6 Tbsp. unsalted butter, melted
1-1/3 cups sifted cake flour
1 cup plus 2 Tbsp. granulated sugar
1/3 cup regular unsweetened cocoa powder
1/2 tsp. baking soda
3/4 cup water
1 egg "large"
1 tsp. vanilla
Adjust rack to center of oven; preheat oven to 375 degrees F. Line the cups of a 12-cup standard muffin tin with paper liners. Melt the butter and cool it until it is warm (it should not be hot when it's added to the batter).
Into a medium bowl, sift together cake flour, sugar, cocoa powder, baking soda, and salt. Add water, egg, vanilla, and melted butter. With hand-held electric mixer at lowest speed (or with whisk), beat ingredients just until combined. Scrape bottom and sides of bowl with rubber spatula. Beat at a low speed for 30 seconds. For ease in handling, transfer batter (which will be on the thin side) to a pitcher of about 4-cup capacity (batter will not fill pitcher).
Divide batter evenly among lined muffin cups; cups will be about two-thirds to three-quarters full. Bake in preheated oven for 14 to 19 minutes, turning pan back-to-front about halfway during baking time. Cupcakes are done when a toothpick inserted into the center of a corner cupcake (that is, a cupcake on one of the corners of the pan) emerges with only a couple of moist crumbs still clinging to it. Do not over bake!
Remove pan to cooling rack. Immediately but gently remove cupcakes from pan to cooling rack (I use a small sharp knife to loosen the cupcakes from the top of the pan, if required, then I gently pry each from the cup with a fork. You'll leave the cupcakes in their paper linings, of course). Cool the cupcakes completely before frosting.
3 Tbsp. unsalted butter, softened
1-3/4 cups confectioners' sugar
3 Tbsp. regular unsweetened cocoa powder
Few grains salt
2 to 3 Tbsp. milk
Add 1 cup of confectioners' sugar, the cocoa and salt and 1 Tbsp. milk. With hand-held electric mixer, beat at medium speed until combined. Scrape bowl and beater(s) with rubber spatula. Gradually add remaining confectioners' sugar and remaining milk, beating at medium speed to incorporate, then increasing speed to high and beating until fluffy. Frosting should have a good spreading consistency.