1. Preheat oven to 350°.
2. To prepare cake, weigh or lightly spoon flour into dry measuring cups, and level with a knife. Combine flour and next 3 ingredients (through salt) in a medium bowl, stirring with a whisk. Add carrot, tossing to combine.
3. Place granulated sugar, brown sugar, and 6 tablespoons butter in a large bowl. Beat with a mixer at medium speed until combined. Add eggs, 1 at a time, beating well after each addition. Stir in 1 teaspoon vanilla. Add flour mixture and buttermilk alternately to sugar mixture, beginning and ending with flour mixture. Spread batter into a 13 x 9-inch metal baking pan coated with cooking spray. Bake at 350° for 28 minutes or until a wooden pick inserted in center comes out clean. Cool cake completely on a wire rack.
4. To prepare frosting, place cream cheese and next 4 ingredients (through salt) in a medium bowl. Beat with a mixer at medium speed until fluffy. Gradually add powdered sugar, beating at medium speed until combined (don't overbeat). Spread frosting evenly over top of cake.