Preheat the oven to 400°
Season both sides of the chicken with ½ tsp black pepper, ½ tsp salt, and paprika.
In an oven-safe large skillet over medium-high heat. Add 2 tbsp of butter and let it melt. Then add chicken and cook on each side for 5-7 minutes or until cooked through. Remove and set aside.
In a large pot of boiling water, cook carrots and potatoes until fork tender. Drain and set aside.
In the same skillet, you cooked the chicken. Add 4 tbsp of butter and mushrooms, onions, and garlic. Saute until soft and tender—about 5 minutes.
Add rosemary, thyme, ½ tsp salt, ½ tsp black pepper, and flour in a small bowl. Whisk and add to vegetables in skillet.
In a medium bowl, add chicken broth and milk. Mix until well combined and pour into skillet. Then mix until well combined. Cook until the sauce begins to thicken. Then add in carrots, potatoes, peas, and chicken. Stir until well combined and remove from heat.
Roll out the puff pastry until it is large enough to cover the skillet with about ½-inch extra on each side. Place it on the chicken pot pie mixture and slice four slits in the center to allow steam to escape.
In a small bowl, add egg and water. Then whisk until well combined. Brush the mixture on the puff pastry.
Bake for 30-35 or until the puff pastry turns golden brown.