Preheat your oven to 350°F.
In a large skillet, heat the olive oil over medium heat. Add the onion and celery and cook for about 3 minutes until the onion is soft and translucent.
Add the chopped chicken to the skillet and stir. Season with salt, pepper, red pepper flakes, and thyme. Cook for another 5 minutes until the chicken is no longer pink.
Add the frozen peas and carrots. Stir in the potatoes and cook for another 5 minutes. Add the cream of mushroom soup and chicken broth. Stir in the parsley.
Cover the skillet with the puff pastry sheet then tuck in the edges. Brush with egg wash, then cut a couple slits in the center of the pie.
Bake for 45 minutes or until the crust is golden brown. Slice and serve warm.