Combine the chicken broth, ¼ teaspoon Kosher salt, 1 teaspoon coarsely ground pepper and thyme sprigs in a large skillet or sauté pan. Bring to a boil then add the chicken breasts making sure they fit in a single layer. Reduce the heat to simmer and cook for 15 minutes, turning once. Turn off the heat and cover the pan. Allow the chicken to rest in the hot broth and finish cooking for about 15 more minutes. Remove the chicken and cut into bite sized pieces, set aside. Remove the thyme sprigs and discard. Set the broth aside until ready to use in the white sauce.

Meanwhile, place the potatoes in a large saucepan and cover with water by one inch. Add 1 tablespoon salt to the water and bring to a boil on high heat. Reduce the heat to medium, maintaining a strong simmer and cook until the potatoes tender, about 15 - 20 minutes. Drain very well. Once dry, place the potatoes back in the pot and add 5 tablespoons butter and salt and pepper to taste. Gently toss the potatoes to cover with the butter. Set aside.

In the large skillet used to poach the chicken, add 1 tablespoon butter and heat on medium-high. Add the chopped mushrooms, season with salt and pepper and cook, stirring occasionally, until nicely browned, about 10 minutes. Add the cooked mushrooms to the white sauce.

Put the skillet back on the heat and melt 1 tablespoon of butter. Add the carrots, onions, and celery and sauté until the onion is soft. Add the vegetables to the white sauce, then add the frozen peas and diced chicken to the sauce. Gently stir to combine.

Pour the chicken and vegetable mixture back into the skillet. Spoon the potatoes over the surface of the filling. Brush the potatoes with the beaten egg and bake for about 20-25 minutes or until the sauce is bubbling and hot and the potatoes are lightly browned.

Allow the casserole to rest 5-10 minutes before serving.

Store leftovers in an airtight container in the refrigerator.