In a large bowl, with an electric mixer on medium speed, beat butter and sugar until smooth. Add egg yolks, whole egg, rum, and vanilla and beat until well blended.
In another bowl, mix 2 cups flour, cornstarch, and baking powder. Stir or beat into butter mixture until well blended. Divide dough in half. Flatten each portion 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 or cooking parchment-lined 12- by 15-inch baking sheets. Gather excess dough into a ball, reroll, and cut out remaining cookies.
Bake in a 350° oven until cookie edges are golden, about 10 minutes; if baking more than one pan at a time, switch pan 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 show at the edges, then roll edges in coconut.
Nutritional analysis is per sandwich cookie.