Preheat oven to 350 degrees. Line two standard 12-cup muffin pans with paper liners; set aside.
In a medium bowl, whisk together flour, baking powder, and salt; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat butter and 2 cups sugar until light and fluffy, 3 to 4 minutes, scraping down sides of the bowl as needed. Beat in vanilla. With mixer on low speed, add flour mixture in three parts, alternating with the milk and beginning and ending with the flour; beat until just combined. Transfer mixture to a large bowl; set aside.
In the clean bowl of an electric mixer fitted with the whisk attachment, beat egg whites on low speed until foamy. With mixer running, gradually add remaining 1/4 cup sugar; beat on high speed until stiff, glossy peaks form, about 4 minutes. Do not overbeat. Gently fold a third of the egg-white mixture into the butter-flour mixture until combined. Gently fold in remaining egg-white mixture; stir in shredded coconut.
Divide batter evenly among the muffin cups, filling each with a heaping 1/4 cup batter. Bake, rotating pans halfway through, until the cupcakes are golden brown and a cake tester inserted in the center of a cupcake comes out clean, 20 to 25 minutes. Transfer pans to a wire rack. Invert cupcakes onto rack; then reinvert and let cool completely, top sides up. Frost cupcakes with seven-minute frosting, swirling to cover. Garnish with fresh coconut just before serving.
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