Prepare the Filling: In a large skillet over medium heat, melt 1 tbsp butter. Add 3/4 cup diced yellow onion and sauté until translucent. Then, add 2 large cubed chicken breasts. Season with salt & pepper and cook until the chicken is well-cooked and golden brown, about 5-7 minutes.
Combine the Remaining Ingredients: In a separate pot, combine 1 cup mixed frozen carrots & peas, 1 cup frozen corn, 1 cup frozen green beans, 3/4 cup diced celery, 3 cups chicken broth, 1 can cream of mushroom soup, 1 can cream of celery soup, 1/2 cup milk, and 1 large cubed russet potato. Stir occasionally as you bring the mixture to a gentle simmer.
Add Chicken to the Vegetable Mixture: Once the vegetables are tender, add the cooked chicken and onions from the skillet to this mixture. In a small bowl, combine the 1 packet McCormick chicken gravy with cold water, mixing it well until smooth. Stir this into the filling, adjusting the thickness with more broth if necessary.
Let it Simmer: Allow the filling to simmer until it thickens, about 10 minutes, stirring often to avoid sticking.
Prepare the Crust: In a mixing bowl, combine your preferred pie crust ingredients (flour, fat, and salt) to create a dough. Divide the dough into two halves, rolling them out to fit your pie dish.
Assemble the Pie: Preheat your oven to 350°F. Spoon the thick filling into a pie dish, cover it with the rolled-out crust, and trim any excess. Brush the top with a mixture of beaten egg and melted butter for a golden finish.
Bake: Place the pie in the preheated oven and bake for about 40 minutes, or until the crust is golden brown and the filling is bubbling.