Preheat oven to 350°.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, 1 3/4 cups sugar, cocoa, baking powder, and salt in a large bowl, stirring well with a whisk. Place butter and chocolate in a microwave-safe bowl; microwave at HIGH for 45 seconds or until melted, stirring once. Combine milk, 1/2 teaspoon vanilla, and eggs, stirring well with a whisk. Add chocolate mixture and milk mixture to flour mixture; beat with a mixer at medium speed until blended. Spoon batter evenly into a 13 x 9-inch baking pan coated with cooking spray.
Place remaining 1/4 cup sugar, remaining 1/4 teaspoon vanilla, cheese, and egg white in a medium bowl; beat at medium speed until well blended using clean, dry beaters. Drizzle cheese mixture evenly over chocolate mixture; swirl batters together using the tip of a knife. Bake at 350° for 30 minutes or until batter begins to pull away from sides of pan. Cool completely on a wire rack.