FOR THE CAKE
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)

FOR THE FROSTING
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

EQUIPMENT
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)