Preheat your oven to 400 degrees F and prepare a 9-inch casserole dish with a light coating of non-stick spray.
In a large skillet over medium heat, warm the olive oil. Add the sliced mushrooms, and sauté for about 5-7 minutes until they are golden and tender.
Stir in the carrots, celery, thyme, salt, and pepper. Cook for an additional 3-4 minutes until the vegetables soften.
Sprinkle the whole wheat flour over the vegetable mixture and cook for 1-2 minutes, stirring well to combine.
Gradually whisk in the oat milk, stirring consistently until the mixture thickens, about 4-6 minutes. Add the shredded chicken, peas, and pearl onions, and mix until well combined.
Transfer the chicken mixture into the prepared casserole dish. Roll out the pie crust over the top, trimming the excess and gently pressing the edges to seal.
Brush the crust with the egg wash, creating a shiny glaze, and make several slits on top to allow steam to escape.
Bake in the preheated oven for about 30-35 minutes, or until the crust is golden and the filling is bubbling. Let it cool slightly before serving.