Heat the olive oil in a large skillet over medium-high heat. Add the ground beef and cook, breaking it into small crumbles, until no pink remains, about 6 to 8 minutes. Drain off excess fat if needed.
Add the onion and bell pepper. Cook until softened, about 4 to 5 minutes. Stir in the garlic, paprika, cumin, oregano, optional cayenne, salt, and black pepper; cook for 30 seconds, just until fragrant.
Remove the skillet from the heat. Stir in the olives and chopped hard-boiled eggs. Transfer the filling to a bowl or shallow pan and let it cool until no longer warm; a cooled filling helps keep the pastry crisp and easy to seal.
Heat the oven to 375°F (190°C). Line a large baking sheet with parchment paper. Whisk the egg and water together in a small bowl to make an egg wash.
Place about 2 tablespoons (30 mL) of cooled filling slightly off-center on each pastry disc. Avoid overfilling. Fold each disc over into a half-moon, press out any trapped air, and firmly crimp the edges with a fork. Arrange the empanadas on the prepared baking sheet.
Brush the tops lightly with egg wash. Bake until deeply golden and crisp, about 20 to 23 minutes. Let stand for 5 minutes before serving, as the filling will be hot.