Position an oven rack in the center of the oven and preheat the oven to 350 degrees F. Lightly butter two 9-by-3-inch round cake pans (springform pans are fine) and line the bottoms with waxed or parchment paper. Dust the insides with flour and tap out the excess.
To make the cake: In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. In a glass measuring cup, mix together the buttermilk and vanilla. In a large bowl, beat the brown sugar and butter together with an electric mixer set on high speed until light and fluffy, about 3 minutes. One at a time, beat in the eggs and yolks, beating well after each addition. With the mixer on low speed, add the flour mixture in thirds, alternating with the buttermilk mixture in 2 equal additions, beating until smooth, and scraping down the sides of the bowl as needed. Fold in the chocolate chips. Divide the batter between the prepared pans and smooth the tops.
Bake until a wooden toothpick in the centers of the cakes comes out clean, about 40 minutes. Let cool on wire cooling racks for 10 minutes. Run a dinner knife around the insides of the pans to release the cakes, invert onto the racks, and remove the pans and paper. Turn the cakes right side up on the racks and let cool completely.
Place a dab of frosting on a serving plate. Place 1 cake layer, flat side up, on the plate. Using a metal icing spatula, spread it with 1 cup of the frosting. Place the second cake layer, flat side down, on top. Frost the top and sides of the cake with the remaining frosting. (The cake is best the day it is made.)
Put the chocolate in a medium heatproof bowl. Bring the cream to a boil in a medium saucepan over medium heat, taking care that it doesn't boil over. Remove from the heat, add the butter, espresso powder, and vanilla, and stir until the butter melts. Pour over the milk chocolate. Let stand until the chocolate softens, about 1 minute.
Whisk until the chocolate is completely melted and smooth. Cover the bowl with plastic wrap. Refrigerate until the mixture is as thick as chocolate pudding, about 2 hours.
Beat the chocolate mixture with an electric mixer set on high speed, just until it becomes lighter in color and spreadable, about 10 seconds. Be very careful not to overbeat, or it will separate. (In that case, melt the mixture in a stainless-steel bowl set over a saucepan of simmering water. Cover and refrigerate until thickened, and try again.) Makes enough for one 9-inch double-layer cake.
NotesThis icing is quick, smooth, and creamy. I love it on the Chocolate Chip Layer Cake. The mocha flavor is very mild and gives just the right touch to the cake. Even though it uses milk chocolate, it is not as sweet as you may think. I usually choose dark chocolate over milk chocolate, but this frosting is fantastic!