Preheat oven to 350 degrees, with racks in upper and lower thirds. Cut two 9-inch rounds of parchment paper to line the bottoms of two 9-inch (2 inches deep) round cake pans. Butter and line pans. Butter linings. Dust pans with flour, tapping out excess. Set aside.

Sift together flour, sugar, baking soda, baking powder, and salt into the bowl of an electric mixer. Return bowl to mixer; fit mixer with the paddle attachment. Add eggs, yolk, water, buttermilk, oil, and vanilla. Mix on low speed until mixture is smooth, about 3 minutes.

Divide batter evenly between prepared pans. Bake, switching positions of pans and rotating halfway through, until a cake tester inserted into centers comes out clean, about 42 minutes. Let cool in pans on wire racks 30 minutes. Run a thin knife around edges of cakes to loosen. Unmold; peel off parchment. Let cool completely, top sides up, on racks.

Trim tops of cake layers with a long serrated knife to make them level. Using a small offset spatula, evenly cover top of one cake layer with 1 1/2 cups frosting; spread frosting so that it extends just beyond edges of cake.

Place second cake layer, cut side down, on top; press gently to make it level. Frost top and sides of assembled cake with remaining frosting.