1. Make cake: Preheat oven to 350°F. Generously grease and flour 2 8-inch round cake pans that are at least 2 inches deep. Line bottom of pans with parchment; grease parchment. Lightly beat eggs, vanilla and 1/4 cup half-and-half.
2. In a large bowl, using an electric mixer on medium-low speed, mix flour, sugar, baking powder and salt. Add butter and remaining 3/4 cup half-and-half. Beat on high speed for 1 1/2 minutes. Scrape down sides of bowl. Add egg mixture in 2 batches, mixing for about 10 seconds at low speed after each addition. (Don't worry if the batter looks a bit curdled.) Divide batter evenly between pans and bake, rotating pans halfway through, until a toothpick inserted in center of a cake comes out clean, 35 to 40 minutes.
3. Let cakes cool in pans on a wire rack for 30 minutes. Run a paring knife around inside edges of pan; turn cakes out onto rack to cool completely.
4. Make frosting: Put chocolate, butter and espresso powder in a metal bowl set over a saucepan filled with 1 inch of simmering water (do not let water boil or allow bottom of bowl to touch water). Cook over medium-low heat, stirring often, until butter and chocolate have melted, espresso powder has dissolved and mixture is smooth. Remove pot from heat. In a large bowl, whisk confectioners' sugar, half-and-half and vanilla together until smooth. Beat chocolate mixture into sugar mixture until glossy.
5. Assemble cake: Level each cake layer with a serrated knife, if desired (see "Frost Like a Pro," below). Place 1 cake layer, bottom side up, on a serving plate. Spread top evenly with about 3/4 cup frosting. Top with second cake layer, bottom side up. Spread remaining frosting evenly over top and sides of cake.
Frost Like a Pro
Follow these simple steps and people might think you bought your dessert at a bakery.
Step 1. Level it.
If your cake has a domed top, use a serrated knife to cut off the dome, making top as level as possible. Flip the bottom layer over so it rests on the cut side before frosting.
Step 2. Frost the middle.
Spread the frosting evenly on the first layer using an offset spatula. Don't worry about making it look perfect, as long as it's even. Be careful not to pull crumbs into the frosting. Place the second layer on top and gently press down.
Step 3. Finish it.
Spread the frosting over the top and sides. To make pretty whorls, run the back of a tablespoon over the frosting in a swirling motion.