MAKE THE DOUGH: In a food processor, pulse the flour with the sugar and salt. Add the butter and pulse until the mixture resembles coarse meal. Beat the eggs with the wine and vinegar and drizzle over the flour mixture. Pulse until the dough just comes together. On a lightly floured work surface, gently knead the dough until smooth. Wrap the dough in plastic and refrigerate until firm, about 1 hour.
PREPARE THE FILLING: In a medium skillet, heat the oil. Add the achiote seeds and cook over moderately high heat until the seeds darken and the oil is orange, about 1 minute. Discard the seeds. Add the ground beef to the skillet and cook, breaking up the meat with a wooden spoon, until no pink remains, about 3 minutes. Add the garlic, onion and bell pepper and cook over moderate heat until the onion is softened, about 5 minutes. Add the tomato, tomato paste and chicken stock and simmer over moderate heat until the liquid has nearly evaporated, about 3 minutes. Stir in the cilantro and season with salt and pepper. Let cool.
On a generously floured work surface, roll out the dough 1/8 inch thick. With a 3-inch round biscuit cutter, stamp out as many rounds as possible (you should have about 24). Reroll the dough scraps and stamp out additional rounds if possible. Brush the excess flour off the rounds. Working with 1 round at a time and keeping the rest covered with plastic wrap, form the empanadas: Spoon 2 teaspoons of the filling on one side of the dough round. Fold the dough over to enclose the filling and crimp the edges with a fork to seal. Cover with plastic wrap while you form the remaining empanadas.
Preheat the oven to 350°. In a deep skillet, heat 1/2 inch of oil to 350°. Fry 4 empanadas at a time, turning once, until browned and crisp, 2 minutes. Drain on paper towels and transfer to a baking sheet. When all of the empanadas have been fried, reheat them in the oven and serve.
The filled, uncooked empanadas can be frozen on a baking sheet, then transferred to an airtight bag. Bake them in a 350° oven without thawing.
Alternatively, you can bake the empanadas: Preheat the oven to 350°. Beat 1 egg with 1 tablespoon of milk. Evenly space the uncooked empanadas on 2 baking sheets lined with parchment paper and brush with the egg wash. Bake in the upper and lower thirds of the oven for about 25 minutes, until golden brown.