Preheat an oven to 400ºF and heat a large Dutch oven or skillet over medium heat with the olive oil. Once hot, add the onion, celery, and carrots. Season with salt and pepper and cook, stirring occasionally, for 6-8 minutes until the veggies begin to soften.
Add the garlic powder, nutmeg, thyme, and chicken broth and cook until very little liquid remains, about another 3-4 minutes.
Turn off the heat and add the chicken, frozen peas, grated cheese, and milk. Stir everything together and press into the pan in an even layer.
Add the pie crust on top, brush with a splash of heavy cream or milk, cut small slits in the center of the crust to vent any steam, and bake for 25-35 minutes until golden brown. You can use a paper towel to blot any leaks and foil to cover any parts of the pie browning too quickly.