For the filling:
2 cups peeled and medium-dice russet potatoes
2 tablespoons vegetable oil
1 pound ground chicken, preferably dark meat
1 cup small diced yellow onion
1 tablespoon smoked paprika
1 teaspoon ground cumin
3 cloves garlic, minced
1/2 cup dry white wine
1 cup low-sodium chicken broth
1/2 cup coarsely chopped fresh parsley leaves
1/3 cup diced green olives
1/3 cup capers
For the empanada dough:
3 cups all-purpose flour, plus more for dusting
2 teaspoons kosher salt
2 teaspoons granulated sugar
1/2 cup vegetable shortening or lard
1 large egg
1/2 cup ice water
1 tablespoon distilled white vinegar
2 large eggs
For the filling: Bring a large saucepan of salted water to a boil over medium-high heat. Add the potatoes and cook until just tender, 5 to 10 minutes. Drain.
Heat the oil in a Dutch oven or large, heavy-bottomed pot over medium-high heat. Add the chicken and a liberal pinch of salt and cook until browned, giving it an occasional stir to ensure it cooks evenly, 4 to 5 minutes. Add the onion, paprika, cumin, and another good pinch of salt to the browned chicken and continue cooking until the onions are translucent and aromatic, 3 to 4 minutes. Add the garlic and cook for 1 minute.
Add the wine, scrape the bottom of the pan with a wooden spoon to release the browned bits, and cook until the wine is reduced by half. Stir in the cooked potatoes and chicken broth, and simmer for 15 minutes more.
Remove from the heat and stir in the olives, capers, and parsley. Transfer to a large bowl and refrigerate until cold, at least 1 hour. Meanwhile, make the empanada dough.
For the empanada dough: Place 3 cups of the flour, salt, and sugar in a food processor fitted with a blade attachment and pulse until evenly combined. Cut the shortening into small chunks, add it to the food processor, and pulse until the mixture has the consistency of sand.
Whisk the egg, water, and vinegar in a small bowl until combined. Turn on the food processor and slowly pour in the egg mixture. Continue mixing until the mixture resembles coarse crumbs.
Empty mixture onto a lightly floured work surface and use your hands to shape it into a rough ball. Using the heels of your hands, gently knead the dough into a smooth, elastic ball. Wrap the dough in plastic wrap and refrigerate for 1 hour.
To finish: Lightly flour a clean, large work surface and a rolling pin. Roll the dough out to about 1/8-inch th