To clarify butter, cut butter into chunks and place in a heavy, medium-size saucepan. Melt over moderately-low heat, 5 to 7 minutes. Remove from heat; let stand 10 minutes. Skim off the white surface foam and discard. Tilt pan slightly, and spoon out clear liquefied butter into clean storage container. Refrigerate for up to 1 month.
Heat oven to 350°F. Film inside of madeleine molds with some of the clarified butter, using soft pastry brush. Let stand 5 minutes, until set. Dust inside of each shell with all-purpose flour. Tap out excess (should be very thin coating).
Onto sheet of waxed paper, sift together cake flour, cocoa, baking powder and salt. In small bowl, toss chocolate chips with 1 teaspoon of sifted flour mixture to coat chips; set aside.
Using electric hand mixer, beat butter in medium-size bowl on moderate speed until creamy, 2 minutes. Add superfine sugar; beat 1 minute. Beat in eggs, one at a time, beating for 30 to 45 seconds after each addition. Scrape down sides of bowl frequently with rubber spatula to keep batter even textured. Blend in vanilla extract and melted bittersweet chocolate. At this point, mixture will look slightly curdled.
On low speed, add sifted flour mixture in 2 additions, mixing just until particles of flour and cocoa are absorbed. Batter will be thick, dense and creamy-textured. Stir in chocolate chips. Let batter stand for 3 minutes. Fill prepared molds with batter, dividing evenly and mounding batter slightly in center of each shell.
Bake in 350°F oven for 12 minutes or until plump, set and wooden pick inserted in the middle of a madeleine withdraws clean. Underbaked madeleines will be gummy. Let madeleines cool in pans on racks 1 minute. With tip of flexible palette knife, prod them out of pan at one long side. Place ridged side up, on racks. Let cool.
Sift together powdered sugar and cocoa into small bowl. Whisk to blend well.
Just before serving, resift powdered sugar mixture on top. Serve madeleines when fresh, within 1 to 2 hours of baking.
You can use vanilla-scented superfine sugar in place of plain superfine sugar in the batter.
To make vanilla-scented sugar: Split open sides of 2 moist vanilla beans. Pour a third of 3 pounds superfine sugar in large storage container; drop in one split bean. Fill with half the remaining sugar; drop in remaining bean. Add rest of sugar. Cover tightly. Stir after 2 days. Let sit another 2 days before using.
Eliminate cocoa powder, chocolate chips. Decrease cake flour to 1/4 cup. Add 1/2 cup plus 2 tablespoons unsifted bleached all-purpose flour. Decrease butter to 6 tablespoons. Decrease eggs to 2. Decrease vanilla to 1 teaspoon; add seed scrapings from 1/4 vanilla bean, and 1/8 teaspoon nutmeg. Replace melted bittersweet chocolate with 2 teaspoons heavy cream. Follow directions. Sift with confectioners sugar when cooled.