Step 1: Heat olive oil in a large skillet or pot over medium-high heat. Add ground beef and cook, breaking it up with a spoon, until browned. Drain any excess fat.
Step 2: Add chopped onion, diced carrots, and diced potatoes to the skillet with the beef. Sauté for 8-10 minutes, or until vegetables begin to soften.
Step 3: Stir in minced garlic, dried thyme, dried rosemary, salt, and black pepper. Cook for 1 minute until fragrant.
Step 4: Sprinkle flour over the beef and vegetable mixture, stirring constantly for 1-2 minutes to cook out the raw flour taste.
Step 5: Gradually pour in the beef broth, stirring continuously to prevent lumps. Bring the mixture to a simmer and cook for 5-7 minutes, or until the sauce thickens. Stir in the frozen peas and remove from heat. Let the filling cool completely.
Step 6: Preheat your oven to 400°F (200°C). Line two baking sheets with parchment paper.
Step 7: On a lightly floured surface, unroll one sheet of thawed puff pastry. Using a 4-5 inch round cookie cutter or a knife, cut out 3-4 circles from each pastry sheet. You may re-roll scraps once to cut more circles.
Step 8: Place a spoonful (about 1/4 to 1/3 cup) of the cooled beef filling onto one half of each pastry circle, leaving a small border around the edge.
Step 9: Fold the other half of the pastry over the filling, creating a half-moon shape. Press the edges firmly with your fingers, then crimp with a fork to seal completely.
Step 10: Carefully transfer the sealed meat pies to the prepared baking sheets. Using a sharp knife, make 2-3 small slits on top of each pie to allow steam to escape.
Step 11: Brush the tops of the pies with the egg wash. Bake for 20-25 minutes, or until the pastry is golden brown and puffed.
Step 12: Let the crispy meat pies cool for a few minutes before serving. Enjoy warm!