Preheat the oven to 375 degrees. Lightly grease a 9-inch pie dish with cooking spray. Set aside.
Heat a cast-iron skillet or dutch oven over medium-high heat. Add the butter to the skillet. Add the onions, mushrooms, and garlic. Cook until the mushrooms and onions begin to brown, stirring occasionally, about 7-10 minutes. Add the carrots, celery, salt, and pepper. Cook for another 3-5 minutes, until carrots begin to soften.
Pour the almond milk into the mixture, adding a little at a time and stirring constantly. Sprinkle the kodiak cakes mix into the mixture and stir. Continue to let the mixture bubble and thicken (about 5 minutes), bringing it to a low boil. Taste test and add additional seasonings.
Add the cooked chicken, thyme and peas to the mixture. Stir.
Scoop the pie mixture into the greased pie dish.
Place the rolled out the pie dough over the pie dish. Place on top of mixture, trimming the overhang to the edge of the dish. Gently press the edges with a fork or your fingers. Cut a few slits in top of crust or poke holes with a fork to let the steam escape. Whisk 1 egg together and brush the top of the crust with the egg wash.
Bake until hot and bubbly on the inside and the crust is deeply golden, about 35-40 minutes. Let rest a few minutes. Serve hot.