1. Preheat oven to 350°.
2. To prepare cake, coat bottom of a 13 x 9-inch baking pan and an 8-inch square baking pan with cooking spray (do not coat sides of pans); line bottoms of pans with wax paper. Coat wax paper with cooking spray; dust pans with 1 1/2 tablespoons flour. Set aside.
3. Combine 3/4 cup butter and sour cream in a large bowl, stirring with a whisk until well blended. Add granulated sugar and 1 tablespoon vanilla. Beat with a mixer at medium speed 3 minutes or until well blended. Add eggs and egg whites, 1 at a time, to sugar mixture, beating well after each addition.
4. Weigh or lightly spoon 13.5 ounces flour (about 3 cups) into dry measuring cups; level with a knife. Combine 13.5 ounces flour, baking soda, and 3/4 teaspoon salt in a large bowl, stirring with a whisk. Add flour mixture and buttermilk alternately to sugar mixture, beginning and ending with flour mixture.
5. Pour about two-thirds of batter into 13 x 9-inch pan; pour remaining batter into 8-inch square pan. Sharply tap pans once on counter to remove air bubbles. Bake at 350° for 30 minutes or until a wooden pick inserted in center comes out clean. Cool in pans on a wire rack for 10 minutes. Remove from pans; cool completely on wire racks.
6. To prepare frosting, place cream cheese, 6 tablespoons butter, and 1 1/2 teaspoons vanilla in a large bowl; beat with a mixer at medium speed until fluffy. Add cocoa, milk, and 1/8 teaspoon salt; beat at low speed until blended. Gradually add powdered sugar, beating until well combined.
7. To prepare baby cake, cut 2 (4-inch) circles out of 8-inch square cake. Place 1 cake layer on a small plate; spread with 3 tablespoons frosting. Top with remaining cake layer. Spread 3/4 cup plus 1 tablespoon frosting over top and sides of cake.
8. To prepare adult cake, place 13 x 9-inch cake on a serving platter; spread 2 cups frosting on top and sides of cake. Store cakes loosely covered in refrigerator.