Chocolate Cake with Chocolate Frosting

This is a great old-fashioned chocolate cake.

1 cup boiling-hot water
3/4 cup unsweetened cocoa powder
1/2 cup half and half or whole milk

2 sticks (1 cup) butter, room temperature
2 cups packed dark brown sugar
4 large eggs, room temperature
2 teaspoons vanilla

2 cups all-purpose flour
1 1/2 teaspoons baking soda
1/2 teaspoon salt

Make cake layers:
Put oven rack in middle position and preheat oven to 350 Butter 2 (9- by 2-inch) round cake pans and line bottom of each with a round of wax paper. Butter paper and dust pans with flour, knocking out excess.

Whisk together hot water and cocoa powder in a bowl until smooth, then whisk in milk and vanilla.

Sift together flour, baking soda, and salt in another bowl.

Beat together butter and brown sugar in a large bowl with an electric mixer at medium-high speed until pale and fluffy, 3 to 5 minutes. Add eggs 1 at a time, beating well after each addition. Reduce speed to low and add flour and cocoa mixtures alternately in batches, beginning and ending with flour mixture (batter may look curdled).

Divide batter between cake pans, smoothing tops. Bake until a wooden pick or skewer comes out clean and edges of cake begin to pull away from sides of pans, 25 to 35 minutes total. Cool layers in pans on racks 10 minutes, then invert onto racks, removing wax paper, and cool completely.

For frosting
2/3 cup whole milk
3 large egg yolks
1 tablespoon plus 1 teaspoon all-purpose flour
1 1/3 cups confectioners sugar
1 teaspoon vanilla
3 sticks (1 1/2 cups) unsalted butter, cut into tablespoon pieces and softened
8 oz milk chocolate, melted and cooled
2 oz unsweetened chocolate, melted and cooled

Make frosting:
Heat milk in a 1- to 1 1/2-quart heavy saucepan over moderate heat until hot. Whisk together yolks, flour, 1/3 cup confectioners sugar, and a pinch of salt in a bowl, then add hot milk in a stream, whisking. Transfer custard to saucepan and bring to a boil over moderate heat, whisking. Reduce heat and simmer, whisking, 2 minutes (mixture will be very thick), then transfer to a large bowl. Cover surface of custard with a buttered round of wax paper and cool completely, about 45 minutes.

Add vanilla and remaining cup confectioners sugar to custard and beat with cleaned beaters at moderate speed until combined well, then increase speed to medium-high and beat in butter, 2 tablespoons at a time, until smooth. Add chocolates and beat until combined well.

Frost cake:
Halve each cake layer horizontally using a long serrated knife. Layer cake, using a heaping 1/2 cup frosting between each layer, then frost top and sides with remaining frosting.