In a large bowl, with an electric mixer on medium speed, beat butter and granulated sugar until smooth. Add egg and beat until well blended. Stir in flour, then beat just until dough comes together. Cover bowl and chill until dough is firm but still pliable, about 30 minutes.
Meanwhile, in a food processor, whirl powdered sugar and almond paste until blended. In a deep bowl, with an electric mixer on high speed (wash and dry beaters after making dough if using the same mixer), beat egg whites until they hold soft peaks. Gently stir in almond-paste mixture until well blended.
On a lightly floured surface, with a floured rolling pin, roll about half the dough at a time to about 1/16 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. Place about 1 teaspoon almond mixture in the center of each round, then fold edges over filling toward the center to form a three-cornered hat shape. With your fingers, pinch corners to seal. Gather excess dough into a ball, reroll, and cut out more cookies; fill and fold. Repeat to roll remaining dough and fill and shape cookies.
Bake cookies in a 325° oven until golden brown, 18 to 20 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. Let cookies cool on sheets for 5 minutes, then use a wide spatula to transfer to racks to cool completely.