1. Preheat oven to 350°.
2 To prepare cake, weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking soda, and 1/2 teaspoon salt, stirring well with a whisk.
3. Place 7 tablespoons butter, 1 cup granulated sugar, and 1/4 cup brown sugar in a mixing bowl; beat at medium-high speed with a mixer until light and fluffy (about 3 minutes). Add eggs, 1 at a time, beating well after each addition. Beat in egg white. Beat in 1 teaspoon vanilla. Add flour mixture alternately with buttermilk, beginning and ending with flour mixture. Fold in 6 tablespoons walnuts. Scrape batter into a 13 x 9–inch pan coated with cooking spray.
4. Bake at 350° for 28 minutes or until a wooden pick inserted in center comes out clean. Cool in pan on a wire rack.
5. To prepare frosting, place 1/2 cup brown sugar, 1/4 cup milk, 2 tablespoons butter, corn syrup, and dash of salt in a saucepan over medium-high heat; bring to a boil, stirring occasionally. Cook 2 minutes. Scrape brown sugar mixture into a bowl. Add remaining 2 tablespoons milk and powdered sugar; beat with a mixer at high speed 2 minutes or until slightly cooled and thick. Beat in 1/2 teaspoon vanilla. Spread frosting in an even layer over cooled cake; sprinkle with 2 tablespoons chopped walnuts. Let the cake stand until the frosting sets; cut into squares.