Make the crust by pulsing gluten free flour, cold butter, and salt in a food processor until pea-sized crumbles form. Add ice water one tablespoon at a time and pulse until dough comes together. Wrap in plastic and refrigerate for 30 minutes.
Preheat oven to 400F. Melt 2 tablespoons butter in a large skillet over medium heat. Add onion, carrots, and celery and cook for 5 minutes until softened.
Sprinkle cornstarch over the vegetables and stir for 1 minute. Slowly pour in chicken broth and heavy cream, stirring constantly. Bring to a simmer and cook until the sauce thickens, about 3 minutes.
Stir in shredded chicken, frozen peas, dried thyme, salt, and pepper. Pour filling into a 9-inch pie dish or cast iron skillet.
Roll out the chilled dough between two sheets of parchment paper to fit the top of the dish. Lay it over the filling and crimp the edges. Cut a few small slits in the top, then brush with beaten egg.
Bake for 30 to 35 minutes until the crust is golden brown and filling is bubbling. Let rest 10 minutes before serving.