Make the Dough: Process the flour, sugar and salt together in a food processor until combined, about 6 seconds. Scatter the butter pieces over the flour mixture and pulse until the mixture resembles coarse crumbs with butter bits no larger than small peas, about 16 pulses.
Transfer the flour mixture to a large mixing bowl. Working with 1/4 cup of water at a time, sprinkle the water over the flour mixture and stir it in using a rubber spatula, pressing the mixture against the side of the bowl to form a dough, until no small bits of flour remain (you may not need to use all of the water).
Turn the dough out onto a clean work surface and divide it into 2 equal pieces. Press each dough half into a cohesive ball, then flatten the ball into a 6-inch disk. Wrap each disk in plastic and refrigerate until firm but not hard, about 2 hours or up to 2 days.
Line 2 baking sheets with parchment paper; set aside. Remove 1 disk of dough from the refrigerator (if refrigerated longer than 2 hours, let sit at room temperature until malleable). Roll the dough out on a lightly floured work surface into an 18-inch circle about 1/8 inch thick. Using a 3-inch round biscuit cutter, cut out 24 rounds and transfer them to the prepared baking sheet. Wrap the baking sheet with plastic wrap and refrigerate. Repeat with the second disk of dough and the second prepared baking sheet.
Make the Filling: Heat the oil in a 12-inch nonstick skillet over medium-high heat until just shimmering. Add the onion and cook until softened, 5 to 7 minutes. Stir in the tomato paste, garlic, oregano, cumin, cloves, and cayenne and cook until fragrant, about 30 seconds. Stir in the beef and cook, breaking up the clumps with a wooden spoon, until no longer pink, about 4 minutes.
Stir in the broth, reduce the heat to low, and simmer until the mixture is moist but not wet, about 8 minutes. Off the heat, stir in the sugar and season with salt and pepper to taste. Transfer mixture to a medium bowl, cover with plastic wrap and refrigerate until completely cool, about 1 hour. Stir in the cheese and continue to chill until needed, up to 3 days.
Assemble the empanadas: Working with the first batch of dough rounds, place about 1 teaspoon of the chilled filling in the center of each dough round and moisten the edge of the dough round with water, using either your finger or a pastry brush. Fold the dough in half over the filling, making a half-moon shape. Pinch the seam along the edge to secure. Using a dinner fork, crimp the sealed edge to secure. Arrange them on a fresh, parchment-lined baking sheet. Wrap the baking sheet tightly with plastic wrap and refrigerate while making a second batch of empanadas using the remaining dough rounds and filling. Follow freezing instructions at this time if making ahead.
To serve: Adjust 2 oven racks to the upper-middle and lower-middle positions and heat the oven to 425 degrees F. Unwrap the empanadas and brush with the egg and bake until golden brown, about 20 minutes, switching and rotating the trays halfway through the baking time. Let cool for 5 minutes before serving.