In a large skillet, heat a tablespoon of oil over medium heat and brown the ground beef until no longer pink, about 5-7 minutes. Drain excess fat, add onions, carrots, and celery, and sauté until soft, about 5 minutes.

Stir in diced potatoes, peas, thyme, and season with salt and pepper. Cover and let simmer until thickened, about 10 minutes.

Preheat oven to 400°F (200°C). Roll out the chilled puff pastry on a floured surface until 1/8 inch thick. Cut into 4-inch squares, and place on a parchment-lined baking sheet.

Spoon a generous amount of filling onto one half of each square, fold the other half over to create a pocket. Press edges to seal and crimp with a fork for decoration.

Place assembled hand pies back on the baking sheet. Whisk together egg and water, brush tops for a golden finish, and cut slits for steam release.

Bake for 24-26 minutes, until pastry is golden and flaky.

Remove from oven and cool on a wire rack for 5-10 minutes. Serve warm, possibly with dipping sauce or salad.