In a skillet over medium heat, add minced beef and cook until browned. Drain excess fat.

Add chopped onion, garlic, carrots, and mushrooms to the skillet. Cook until vegetables are softened, about 5-7 minutes.

Stir in tomato paste, Guinness, Worcestershire sauce, thyme, salt, and pepper. Bring to a simmer and cook for 20 minutes until thickened.

In a bowl, combine flour and salt. Add cold butter and rub into the flour until the mixture resembles breadcrumbs.

Add cold water gradually until the dough comes together. Wrap in plastic wrap and refrigerate for 30 minutes.

Preheat the oven to 200°C (400°F). Roll out the pastry on a floured surface to fit your pie dish.

Place the filling in the pie dish and cover with the rolled pastry. Trim excess pastry and crimp the edges.

Brush the top with beaten egg and cut a few slits for steam to escape.

Bake in the preheated oven for 30-35 minutes or until the pastry is golden brown.

Let the pie cool for a few minutes before serving.