Lightly coat a 9-inch tart tin plate with nonstick vegetable spray. In a bowl, stir together the flour, sugar, and 1/4 cup of the coconut. With fingers or a pastry blender, work in the butter until the mixture looks like coarse meal. Sprinkle on 2 to 3 tablespoons of the egg water and blend together. Form into a disc and wrap in plastic. Refrigerate for at least 1/2 hour. Roll the dough into a 9 1/2-inch round and fit into the tart tin.
Mix together the remaining 2 cups coconut, egg whites, confectioners' sugar and vanilla. Lightly press mixture against the sides of the tin. Chill for 30 minutes then bake in a preheated 375-degree F oven for 8 minutes, until just set.
Key Lime Filling: In a bowl, mix together the yolks, sweetened condensed milk, key lime zest and juice. Pour it into the shell and bake in a preheated 325 degree F oven for 18 minutes and just set. Set on a rack to cool, refrigerate until chilled.