Coat two (9-inch/23cm) round cake pans with cooking spray; line bottoms with wax paper or parchment paper. Coat paper with cooking spray; dust with the 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 degrees F (180 C) for 30 minutes* (see note), 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 or parchment paper; cool completely on wire rack.
*Note: If using non-stick or dark colored cake pans, bake at 325 degrees F (160 C), and bake for 25-27 minutes.
To Prepare the Caramel 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 confectioners' 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, and garnish top of cake with pecan halves (optional). Store cake loosely covered in refrigerator.