In a 2- to 3-quart pan, combine butter and cocoa; stir over medium-low heat until butter is melted and mixture is well blended. Remove from heat and stir in sugar and vanilla. Add egg and egg whites; beat until blended. Stir in flour until well blended.

Spread batter evenly in an oiled 9-inch-square baking pan. Sprinkle walnuts evenly over batter. Bake in a 325-degree oven until brownies feel dry on top (about 35 minutes). Let cool in pan on a rack; then cut into 2 1/4-inch squares. Store airtight.