Preheat oven to 375°F.
In a 9.5-inch cast-iron skillet, cook the beef until no longer pink. If excess grease has accumulated, then drain off with a spoon. Season the beef with ½ tsp salt and pepper, each. Transfer to a separate bowl and set aside.
In the same skillet, heat the oil over medium heat. Add the onions and sauté until soft, about 4 minutes. Add the garlic and thyme and cook for another 30 seconds, stirring often.
Add the carrots, celery, and potatoes, and cook for 10 minutes, stirring often.
Sprinkle the flour on and cook for another 1 minute.
Add the tomato paste, Worcestershire sauce, beef broth, and beer. Bring to a boil, then lower the heat and simmer for 10 minutes, stirring occasionally.
Stir in 1 tsp salt, ½ tsp pepper, and the peas. Stir in the ground beef. Stir until fully combined. Simmer another 5 minutes.
Meanwhile, roll out the dough on a lightly floured surface, to a 10-inch diameter. Remove the skillet from the heat. Carefully place the dough over the top of the skillet. Crimp the edges along the side of the pan.
In a small bowl, combine 1 egg with 1 tbsp water. Stir with a fork to fully combine. Brush the egg wash over the top of the dough. Use a sharp knife to make 3 slits in the center portion of the dough.
Bake for 40 to 50 minutes, until the crust is golden and the inside is bubbly. Remove from the oven and let rest for 5 to 10 minutes.