In a large bowl, with a mixer on medium speed, beat 1 cup butter and sugar until smooth. Add egg yolks, whole egg, rum, and vanilla and beat until well blended.
In a medium bowl, mix 2 cups flour, cornstarch, and baking powder. Stir into butter mixture, then beat until well blended. Divide dough in half, press each half into a disk, wrap in plastic wrap, and freeze until firm, about 30 minutes.
Unwrap dough. On a lightly floured surface, with a floured rolling pin, roll one disk at a time to about 1/8 inch thick. With a floured, 2- to 3-inch round cutter, cut out cookies. Place about 1 inch apart on buttered 12- by 15-inch baking sheets. Gather excess dough into a ball, reroll, and cut out remaining cookies.
Bake in a 350º regular or convection oven until cookie edges just begin to brown, about 10 minutes. If baking two sheets at once in one oven, switch their positions halfway through baking. Let the cookies cool on sheets for 5 minutes, then use a wide spatula to transfer them to racks to cool completely.
Turn half the cookies bottom side up and spread each with about 1 tablespoon dulce de leche. Top with remaining cookies, bottom side down. Place coconut in a shallow bowl. Gently squeeze each sandwich until filling begins to ooze out sides, then roll edges in coconut.
Nutritional analysis per sandwich.