Spray a 9-inch round cake pan with 2-inch sides or a 9-inch springform pan with cooking spray. Line pan with parchment paper; spray with cooking spray.
Melt butter in a small saucepan over medium-low, swirling pan occasionally, until butter is foamy and starting to brown on the bottom, about 5 minutes. (It should smell like hazelnuts.) Scrape browned bits from bottom; remove from heat, and cool 5 minutes.
Whisk together flour, cocoa, brown sugar, baking powder, baking soda, salt, and 1/2 cup granulated sugar in a large bowl until no lumps of brown sugar remain. (Break up lumps with hands, if necessary.)
Whisk together buttermilk, eggs, and vanilla in a medium bowl until no visible bits of egg remain. Add buttermilk mixture to flour mixture, whisking just until blended (a few streaks of flour are fine). Add brown butter and coconut oil, and whisk just until blended.
Pour batter into prepared pan; sprinkle with coconut flakes and remaining 1 tablespoon granulated sugar. Bake at 350°F for 35 minutes or until coconut is golden brown, cake pulls away from the sides of the pan, and a wooden pick inserted into center of cake comes out clean. Cool cake in pan 10 minutes. Remove cake from pan to a wire rack; cool to room temperature.
Dining In (Clarkson Potter, $30).
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