Heat a large Dutch oven or similar deep, heavy-bottomed pan over medium-high heat. Add the oil to the pan. Once hot, add the mushrooms and cook for 8 minutes, until mushrooms are beginning to brown, stirring occasionally. Add the carrots, celery, garlic powder,  salt, and pepper. Cook until the mushrooms have browned more deeply and the carrots begin to soften, about 3 additional minutes.

Sprinkle the flour over the top of the vegetables and cook 2 minutes. Slowly pour in the almondmilk, adding a few splashes at a time, stirring constantly. Bring to a low boil, scraping any brown bits from the bottom of the pan. Continue to let bubble until thickened, about 3 to 5 minutes. Stir in the chicken, peas, onions, and thyme. Spoon the chicken mixture into the prepared pie dish.

Roll the pie dough into a circle large enough to cover your dish. Brush the edges of the pie dish with the egg wash, then lay the dough over the top so that it overhangs the sides. Trim the overhang to a 1/2 inch larger than edge of the dish. Gently press the dough onto the sides of the dish so that it sticks, then brush all over with the remaining egg wash. With a sharp knife, cut 3 slits in the top.

Bake until hot and bubbly on the inside and the crust is deeply golden, about 25 minutes. Let rest a few minutes. Serve hot.