Preheat oven to 350°.
To prepare cake, coat 2 (9-inch) round cake pans with cooking spray; line bottoms with wax paper. Coat wax paper with cooking spray; dust with 1 tablespoon flour.
Beat granulated sugar and 1/2 cup butter at medium speed of a mixer until well-blended (about 5 minutes). Add eggs and egg white, 1 at a time, beating well after each addition. Lightly spoon 2 1/4 cups flour into dry measuring cups; level with a knife. Combine 2 1/4 cups flour, baking powder, and salt; stir well with a whisk. Add flour mixture to sugar mixture alternately with 1 1/4 cups milk, beginning and ending with flour mixture. Stir in 2 teaspoons vanilla.
Pour batter into prepared pans, and 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 pan 10 minutes on a wire rack; remove from pans. Carefully peel off wax paper; cool completely on wire rack.
To prepare frosting, combine brown sugar and next 4 ingredients (brown sugar through salt) in a medium saucepan, and bring to a boil over medium-high heat, stirring constantly. Reduce heat, and simmer until thick (about 5 minutes), stirring occasionally. Remove from heat. Add powdered sugar and 2 1/2 teaspoons vanilla; beat at medium speed of a mixer until smooth and slightly warm. Cool 2 to 3 minutes (frosting will be thin but thickens as it cools).
Place 1 cake layer on a plate; spoon 1/2 cup frosting on to cake layer spreading to cover. Top with remaining cake layer. Frost top and sides of cake. Store cake loosely covered in refrigerator.