Preheat oven to 400°F (200°C). Lightly grease a 12-cup muffin pan with butter or nonstick spray, coating the sides well.
Heat olive oil in a large skillet over medium-high heat. Cook onion for 2-3 minutes until soft, then add garlic and cook for 30 seconds until fragrant.
Add ground beef and cook for 5-7 minutes, breaking it up with a spoon, until fully browned. Drain excess fat if needed.
Add carrot and cook for 2 minutes. Sprinkle flour over the mixture and stir constantly for 1 minute to cook out the raw flour taste.
Pour in beef stock slowly while stirring. Add tomato paste, Worcestershire sauce, thyme, salt, and pepper. Simmer for 5-6 minutes until thickened.
Fold in frozen peas during the last minute of cooking. Remove from heat and let cool for at least 15 minutes before assembling.
Lightly flour your work surface and roll out shortcrust pastry to 1/8 inch thickness. Use a large round cutter to cut circles for the base.
Roll out puff pastry and use a smaller cutter to cut lid circles. Re-roll any scraps as needed but handle the dough as little as possible.
Press each shortcrust circle into a muffin cup, pushing it into the bottom corners and up the sides with no gaps or tears.
Spoon cooled beef filling into each pastry-lined cup, filling almost to the top. Brush the exposed pastry edge lightly with beaten egg.
Place a puff pastry lid over each filled cup and press the edges to seal. Crimp with a fork and cut a tiny slit in the top for steam to escape.
Brush the top of each pie generously with beaten egg. Bake for 22-25 minutes until deeply golden and puffed, rotating the pan halfway through.
Let the pies rest in the pan for 5 minutes before removing. Run a thin knife around the edge to loosen, then lift out onto a wire cooling rack.