1. Preheat oven to 350°.
2. To prepare cake, coat 2 (8-inch) round and 1 (8-inch) square cake pans with cooking spray; dust with 3 tablespoons flour. Weigh or lightly spoon 11.2 ounces flour (about 2 1/2 cups) into dry measuring cups; level with a knife. Combine flour, baking powder, and next 4 ingredients in a large bowl, stirring with a whisk. Combine granulated sugarand next 7 ingredients in a medium bowl; beat well with a mixer at medium speed. Stir in carrot. Add to flour mixture, stirring just until moist.
3. Pour 1 1/3 cups batter into each prepared pan. Bake at 350° for 20 minutes or until a wooden pick inserted in center comes out clean. Run a knife around outside edge of pans. Cool in pans 10 minutes on a wire rack; remove from pans. Cool completely on wire racks.
4. To prepare frosting, combine butter and cream cheese in a large bowl. Beat with a mixer at medium speed until smooth. Gradually beat in powdered sugar and vanilla just until smooth. Cover and refrigerate until ready to use.
5. For baby's cake, cut 2 small layers from square layer, using a (4-inch) round cookie cutter, discarding scraps. Place 1 layer on a small plate; spread with 1/3 cup frosting. Top with remaining layer. Spread 1/2 cup frosting on top and sides of cake.
6. For adult cake, place 1 round cake layer on a plate; spread with 1/2 cup frosting. Top with remaining round layer. Spread 2 1/4 cups frosting on top and sides of cake. Store cakes loosely covered in refrigerator.