In a mixing bowl, combine all-purpose flour and salt. Cut in chilled unsalted butter until the mixture resembles coarse crumbs. Gradually drizzle in ice water, mixing gently until a dough forms. Divide the dough in half, shape into discs, wrap in plastic, and chill in the refrigerator for 30 minutes to let the butter firm up for a flaky crust.

In a skillet over medium heat, sauté chopped onion in a tablespoon of oil until translucent, about 3-4 minutes. Add minced garlic and diced carrot, cooking until softened, around 5 minutes. Add ground beef to the skillet, browning it completely. Drain excess fat and stir in tomato paste, Worcestershire sauce, dried thyme, salt, black pepper, and beef broth, simmering for 10 minutes. Lastly, mix in frozen peas and allow the filling to cool.

Preheat your oven to 400°F (200°C) while rolling out one disc of the chilled pastry dough on a floured surface. Carefully line a pie dish with the rolled dough, letting the excess hang over the edges. Spoon the cooled meat mixture into the crust, then roll out the second dough disc to cover the pie. Seal the edges by crimping them together, ensuring no filling escapes.

Using a sharp knife, cut slits into the top crust to allow steam to escape. Brush the surface with a whisked egg for a beautiful golden shine. Place the meat pie in the preheated oven and bake for 30-35 minutes or until the crust is golden brown and the filling is bubbling.

Once baked, remove the meat pie from the oven and let it cool on a wire rack for at least 10 minutes. This cooling time helps the filling set, making it easier to slice.