Preheat oven to 350°.
To prepare cake, coat a 13 x 9-inch baking pan with cooking spray; line bottom of pan with wax paper. Coat wax paper with cooking spray; set aside.
Place 9 tablespoons butter, brown sugar, and granulated sugar in a large bowl; beat with a mixer at medium speed 5 minutes or until well blended. Add eggs and egg whites, 1 at a time, beating well after each addition until pale and fluffy. Beat in 2 teaspoons of vanilla extract.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking soda, cinnamon, and 1/4 teaspoon salt, stirring with a whisk. Add flour mixture and buttermilk alternately to sugar mixture, beginning and ending with flour mixture; mix after each addition. Stir in carrot. Spoon batter into prepared pan. Sharply tap pan 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 pan 10 minutes on a wire rack; remove from pan. Carefully peel off wax paper; cool completely on wire rack.
To prepare frosting, place the cream cheese, 1/4 cup butter, 2 teaspoons of vanilla, and 1/8 teaspoon salt in a large bowl; beat with a mixer at medium speed until smooth. Gradually add 2 cups powdered sugar, beating at low speed until smooth (do not overbeat). Stir in the remaining 3/4 cup powdered sugar. Cover and chill 30 minutes. Spread frosting over top of cake. Garnish with sprinkles. Store cake loosely covered in refrigerator.