Ingredients

• 3 cups sifted cake flour (not self-rising; sift before measuring)
• 2 teaspoons baking powder
• 3/4 teaspoon baking soda
• 1 teaspoon salt
• 2 1/4 sticks (1 cup plus 2 tablespoons) unsalted butter, softened
• 2 cups sugar
• 4 large eggs, warmed in warm water 5 minutes
• 2 teaspoons pure vanilla extract
• 1 1/2 cups well-stirred plain, whole-milk yogurt (not Greek-style), at room temperature (see Cooks' Notes)
• 3/4 cup sugar
• 3 tablespoons cornstarch
• 1/4 teaspoon salt
• 1 cup boiling-hot water
• 1 tablespoon light corn syrup
• 8 ounces bittersweet chocolate (60% cacao), chopped fine
• 2 ounces unsweetened chocolate (100% cacao), chopped fine
• 1 teaspoon pure vanilla extract
• 3 sticks (1 1/2 cups) unsalted butter, cut into tablespoon pieces and softened
• 3 (8- or 9-inch) round cake pans, 2 inches deep; large sifter or sieve; electric stand mixer (preferred but optional), fitted with paddle attachment; offset metal spatula (for frosting cake)

Instructions

1. Heat oven to 350°F with racks in upper and lower third. Butter cake pans, then line bottoms with rounds of wax paper or parchment and butter the paper. Dust pans with additional flour and knock out excess.
2. Sift together flour, baking powder, baking soda, and salt into a bowl.
3. Beat butter with sugar in a large bowl at medium-high speed (preferably using a stand mixer) until pale and fluffy, 3 to 5 minutes. Add eggs, 1 at a time, beating well after each addition. Add vanilla and beat mixture 5 minutes more.
4. Reduce speed to low, then add flour mixture alternately with yogurt in 3 batches, beginning and ending with flour mixture and mixing just until smooth. Divide batter between pans (about 2 3/4 cups per pan), spreading evenly, then rap each pan on counter once or twice to eliminate air bubbles.
5. Bake layers, switching position of pans and rearranging them in oven halfway through baking time, until cake begins to pull away from side of pan and a wooden pick inserted in center of each comes out clean, 15 to 20 minutes total for 9-inch pans and 20 to 25 minutes total for 8-inch pans.
6. Cool 10 minutes in pans on racks, then run a thin knife around edge of pans and invert layers onto racks to cool completely.
7. Make frosting: Whisk together sugar, cornstarch, and salt in a heavy saucepan, then whisk in boiling-hot water. Bring to a boil over medium heat, whisking, then reduce heat and simmer, whisking constantly, 1 minute (mixture will be thick). Remove pan from heat and whisk in corn syrup. Add chocolates and vanilla, then stir once. Let stand 1 minute and stir again until chocolate is melted and mixture is smooth. Transfer chocolate mixture to a metal bowl and set bowl in a larger bowl of ice water. Stir chocolate mixture until it has cooled to room temperature, about 5 minutes. With an electric mixer, beat in butter, 2 tablespoons at a time, until incorporated and frosting is light and holds soft peaks.
8. Whisk together sugar, cornstarch, and salt in a heavy saucepan, then whisk in boiling-hot water. Bring to a boil over medium heat, whisking, then reduce heat and simmer, whisking constantly, 1 minute (mixture will be thick).
9. Remove pan from heat and whisk in corn syrup. Add chocolates and vanilla, then stir once. Let stand 1 minute and stir again until chocolate is melted and mixture is smooth.
10. Transfer chocolate mixture to a metal bowl and set bowl in a larger bowl of ice water. Stir chocolate mixture until it has cooled to room temperature, about 5 minutes.
11. With an electric mixer, beat in butter, 2 tablespoons at a time, until incorporated and frosting is light and holds soft peaks.
12. Assemble cake: Put a layer, bottom side up, on a cake plate or stand. Spread 1 cup of frosting over top. Cover with another layer, bottom side up, and spread with another cup of frosting. Top with third layer, bottom side up. Spread a thin layer of frosting around sides and over top of cake with offset spatula. (This is called crumb-coating. It tamps down any loose crumbs to keep them out of the top layer of frosting and fills in any crevices.) Chill or briefly freeze cake just until frosting is firm. Spread sides and top of cake with remaining frosting.
13. Put a layer, bottom side up, on a cake plate or stand. Spread 1 cup of frosting over top. Cover with another layer, bottom side up, and spread with another cup of frosting. Top with third layer, bottom side up.
14. Spread a thin layer of frosting around sides and over top of cake with offset spatula. (This is called crumb-coating. It tamps down any loose crumbs to keep them out of the top layer of frosting and fills in any crevices.) Chill or briefly freeze cake just until frosting is firm.
15. Spread sides and top of cake with remaining frosting.