Preheat oven to 325°.
Coat a 12-cup Bundt pan with cooking spray; dust with breadcrumbs.
Melt 4 tablespoons butter in a large saucepan over medium heat. Add brown sugar, and cook 2 minutes, stirring constantly (sugar will not melt). Pour mixture into a large bowl; cool 20 minutes, stirring occasionally.
Add 3 tablespoons butter, cheese, and vanilla to bowl; beat with a mixer at low speed for 1 minute. Beat at high speed 4 minutes (mixture will not be smooth). Add the eggs and egg white, 1 at a time, beating well after each addition.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour and the next 3 ingredients (flour through salt), stirring well with a whisk. Add flour mixture to sugar mixture alternately with milk, beginning and ending with flour mixture; mix after each addition.
Pour batter into prepared pan. Bake at 325° for 1 hour and 5 minutes or until a wooden pick inserted an inch from the edge comes out clean. Cool in pan for 10 minutes on a wire rack; remove from pan. Cool completely on wire rack. Sift powdered sugar over cake.