Preheat oven to 350° F. In the bowl of a food processor fitted with a metal blade, combine the coconut with the egg whites and pulse until well ground. Lightly coat four 4-inch tart pans with cooking spray. Press the coconut into the tart pans using a firm hand, working it over the bottom and up the sides. Bake until the coconut turns golden around the edges and is set in the center, 15 to 20 minutes. Remove from oven and invert the tins immediately on a rack.