Line a baking sheet with a silicone baking mat. Beat egg whites in the bowl of a stand mixer fitted with a whisk attachment until whites are foamy; beat in white sugar and continue beating until egg whites are glossy, fluffy, and hold soft peaks. Sift confectioners’ sugar and ground almonds in a separate bowl and quickly fold the almond mixture into the egg whites, about 30 strokes.
Spoon a small amount of batter into a plastic bag with a small corner cut off and pipe a test disk of batter, about 1 1/2 inches in diameter, onto prepared baking sheet. If the disk of batter holds a peak instead of flattening immediately, gently fold the batter a few more times and retest. When batter is mixed enough to flatten immediately into an even disk, spoon into a pastry bag fitted with a plain round tip. Pipe the batter onto the baking sheet in rounds, leaving space between the disks. Let the piped cookies stand out at room temperature until they form a hard skin on top, about 1 hour.
Preheat oven to 285 degrees F (140 degrees C). Bake cookies until set but not browned, about 10 minutes; let cookies cool completely before filling.