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 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.
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.
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.
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.
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.
