Step 1: Heat olive oil in a large skillet over medium-high heat. Add the ground beef and cook, breaking it apart with a spoon, until thoroughly browned and no pink remains. Drain any excess fat.
Step 2: Reduce heat to medium. Add the finely diced onion and minced garlic to the skillet with the beef. Cook, stirring occasionally, until the onion is softened and translucent, about 5-7 minutes.
Step 3: Stir in the tomato paste, Worcestershire sauce, dried oregano, smoked paprika, salt, and black pepper. Cook for another 2-3 minutes, stirring constantly, to allow the spices to bloom and the tomato paste to deepen in color.
Step 4: Pour in the beef broth and bring the mixture to a simmer. Reduce the heat to low and let it cook for 10-15 minutes, allowing the liquid to mostly evaporate and the flavors to meld beautifully. The filling should be thick and moist, but not watery.
Step 5: Remove the skillet from the heat and let the beef mixture cool slightly. Stir in the shredded cheddar cheese and mozzarella cheese until fully combined and melted from the residual heat. Taste and adjust seasoning if necessary. Set aside to cool completely.
Step 6: Preheat your oven to 400°F (200°C). Line two baking sheets with parchment paper.
Step 7: Lightly flour a clean work surface. Unfold one sheet of thawed puff pastry. Using a rolling pin, gently roll it into a slightly larger square, about 10×10 inches. Cut the pastry into four equal squares (approximately 5×5 inches each). Repeat with the second sheet of puff pastry, yielding a total of eight squares.
Step 8: Spoon about 1/4 cup of the cooled beef and cheese filling onto one half of each pastry square, leaving a small border around the edges. Do not overfill.
Step 9: Lightly brush the edges of each pastry square with the egg wash (1 large egg whisked with 1 tbsp water).
Step 10: Fold the other half of the pastry over the filling to form a triangle or a rectangle. Press the edges firmly together with your fingers. Then, use a fork to crimp the edges, ensuring a tight seal and creating a decorative border.
Step 11: Carefully transfer the assembled hand pies to the prepared baking sheets. Using a small sharp knife, cut a few slits (2-3) on the top of each pie to allow steam to escape. Brush the tops of all the hand pies with the remaining egg wash.
Step 12: Bake for 20-25 minutes, or until the pastry is puffed, golden brown, and cooked through. Keep an eye on them towards the end of the baking time.
Step 13: Remove the hand pies from the oven and let them cool on the baking sheets for a few minutes before transferring them to a wire rack to cool slightly further. Serve warm and enjoy!