Melt ½ cup butter in a heavy bottom saucepan over medium heat. Whisk in the ½ cup of flour and stir until smooth. Continue stirring for 2 minutes bringing the mixture to a boil. Gradually add the chicken broth and 1½ cups milk. Stir until the sauce is thick and bubbly. Turn off the heat and add 1 teaspoon salt, 1 teaspoon pepper and lemon juice. Cover and set aside.
Preheat oven to 350 degrees.
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.