Position a rack in the center of the oven and heat to 350°F. Using a cylinder-shaped pastry brush, butter the inside of a 9- to 9½-inch Gugelhupf mold. Coat with flour, tapping out the excess.
Melt the chocolate in the top part of a double boiler over hot, not simmering, water or in a microwave oven. Remove from the heat and cool, stirring occasionally, until tepid.
Sift the flour, baking powder, and salt together. Mix the milk and vanilla in a liquid measuring cup.
Beat the butter in a large bowl with a handheld electric mixer on high speed until smooth, about 1 minute. On low speed, beat in the confectioners’ sugar, then increase the speed to medium-high and beat until the mixture is very light in color and texture, about 2 minutes. Beat in the egg yolks, one at a time.
Using clean beaters, beat the egg whites on high speed until soft peaks form. Gradually beat in the granulated sugar until the whites form shiny, soft peaks. Stir about one fourth of the whites into the batter. Stir in half of the flour, then half of the milk; repeat. Fold in the remaining whites. Transfer about one third of the batter to a medium bowl and stir in the melted chocolate.
Spread half of the plain batter in the pan. Spoon in the chocolate batter, then top with the remaining plain batter. Zigzag a knife through the batter.
Bake until a toothpick inserted in the center of the cake comes out clean, about 1 hour. Cool in the pan on a wire rack for 10 minutes. Invert onto the rack and cool completely. Before serving, sprinkle with confectioners’ sugar.