In a large bowl, with a mixer on medium speed, beat 3/4 cup butter and granulated sugar until smooth. Add egg and beat until well blended. Stir in 2 cups flour, then beat just until dough comes together.
In a food processor, whirl powdered sugar and almond paste until blended. In a deep bowl, with a mixer on high speed, beat egg whites until they hold soft peaks. Gently stir in almond 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 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° regular or convection oven until golden brown, 18 to 20 minutes. If baking two sheets at once in one oven, switch their positions halfway through baking. Let cookies cool on sheets for 5 minutes, then use a wide spatula to transfer to racks to cool completely.
Nutritional analysis per cookie.