Preheat oven to 350 degrees with rack in lower third. Place almonds in a food processor; process until as fine as possible, about 1 minute. Add confectioners' sugar; process until combined, about 1 minute.
Pass almond mixture through a fine-mesh sieve. Transfer solids in sieve to food processor; grind and sift again, pressing down on clumps. Repeat until less than 2 tablespoons of solids remains in sieve.
Whisk egg whites and granulated sugar by hand to combine. Beat on medium speed (4 on a KitchenAid) 2 minutes. Increase speed to medium-high (6) and beat 2 minutes. Then beat on high (8) 2 minutes more.
Add dry ingredients all at once. Fold with a spatula from bottom of bowl upward, then press flat side of spatula firmly through middle of mixture. Repeat just until batter flows like lava, 35 to 40 complete strokes.
Rest a pastry bag fitted with a 3/8-inch round tip (Ateco #804) inside a glass. Transfer batter to bag; secure top. Dab some batter remaining in bowl onto corners of 2 heavy baking sheets; line with parchment.
With piping tip 1/2 inch above sheet, pipe batter into a 3/4-inch round, then swirl tip off to one side. Repeat, spacing rounds 1 inch apart. Tap sheets firmly against counter 2 or 3 times to release air bubbles.
Bake 1 sheet at a time, rotating halfway through, until risen and just set, 13 minutes. Let cool. Pipe or spread filling on flat sides of half of cookies; top with remaining half. Wrap in plastic and refrigerate.