In a large bowl, mix 2 cups of flour and 1 teaspoon of salt. Add the cold, cubed butter and use a pastry cutter or your fingers to combine until the mixture resembles coarse crumbs. Gradually add ice water, one tablespoon at a time, mixing until the dough comes together. Form the dough into a ball, wrap it in plastic wrap, and refrigerate for at least 30 minutes.

In a large skillet, heat 2 tablespoons of olive oil over medium heat. Season the diced chicken thighs with salt and pepper, then add them to the pan. Cook until lightly browned and cooked through, about 5-7 minutes. Remove the chicken from the skillet and set aside.

In the same skillet, add the diced onions, carrots, and mushrooms. Sauté for 5-7 minutes until they start to soften. Add the minced garlic and cook for another minute until fragrant.

Sprinkle 1/4 cup of flour over the sautéed vegetables and stir to combine. Gradually add 4 cups of chicken broth while continuously stirring to avoid lumps. Bring the mixture to a simmer, and cook for about 5 minutes until thickened.

Stir in the cup of heavy cream, dried thyme, dried rosemary, and cooked chicken. Add the frozen peas, and season with salt and pepper to taste. Remove from heat and let it cool slightly.

While the filling cools, preheat your oven to 400°F (200°C).

Roll out the chilled dough on a floured surface to fit your pie dish. Place the dough in the bottom of the dish, allowing excess dough to hang over the edges. Pour the chicken and mushroom filling into the pie crust.

Roll out the second half of the dough and lay it over the filling. Trim any excess dough and crimp the edges to seal. Cut a few slits in the top for steam to escape. Brush the crust with the beaten egg for a golden finish.

Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown. Let it cool for a few minutes before serving.

Slice the pot pie, serve it warm, and enjoy the cozy flavors!