Position rack in bottom third of oven; preheat to 325°F. Line 8x8x2-inch metal baking pan with foil, pressing foil firmly against pan sides and leaving 2-inch overhang. Coat foil with nonstick spray. Melt butter in medium saucepan over medium heat. Continue cooking until butter stops foaming and browned bits form at bottom of pan, stirring often, about 5 minutes. Remove from heat; immediately add sugar, cocoa, 2 teaspoons water, vanilla, and 1/4 teaspoon (generous) salt. Stir to blend. Let cool 5 minutes (mixture will still be hot). Add eggs to hot mixture 1 at a time, beating vigorously to blend after each addition. When mixture looks thick and shiny, add flour and stir until blended. Beat vigorously 60 strokes. Stir in nuts. Transfer batter to prepared pan.
Bake brownies until toothpick inserted into center comes out almost clean (with a few moist crumbs attached), about 25 minutes. Cool in pan on rack. Using foil overhang, lift brownies from pan. Cut into 4 strips. Cut each strip crosswise into 4 brownies. DO AHEAD Can be made 2 days ahead. Store airtight at room temperature.