Add eggs, one at a time, mixing well after each addition. In a large bowl, whisk together flour, baking powder, baking soda, and salt. With the mixer running, add flour mixture in two batches, mixing well after each addition. Continue mixing until a dough forms. Form dough into a ball and wrap with plastic wrap; refrigerate until firm, at least 4 hours.

Preheat oven to 350 degrees. Line a baking sheet with parchment paper. Roll dough into 1 1/2-inch balls. Place on prepared baking sheet about 2 inches apart. Press down on each dough ball using the tines of a fork, once vertically and once horizontally, to make a crosshatch pattern on the cookies. Sprinkle with sugar and transfer to oven. Bake 14 to 16 minutes, rotating baking sheet once during baking. Let cool completely.