Prepare the crust by pouring flour into a mixing bowl, season with salt and mix to combine. Using a pastry cutter, cut butter into pea-sized pieces. Add water and work flour into dough until fully incorporated. Roll dough out onto a lightly floured surface. Cut into strips and weave into a lattice or leave whole. Cut crust to fit cast-iron skillet you are using and place in refrigerator to chill while you prepare the filling.
Pre-heat oven to 425 degrees F.
Preheat a 10-inch cast-iron skillet over moderate heat. Season chicken with 1 teaspoon salt. Coat surface of pre-heated skillet with oil and fully cook the chicken. Remove chicken from skillet and medium dice when cool enough to touch, set aside to be re-introduced later.
Add onion, celery and carrots to skillet, cover and steam until soft, approximately 10 minutes. Add garlic, season with pepper, thyme, mustard and remaining 1 ¼ teaspoon salt. Saute for an additional two minutes uncovered. Mix in flour, removing any clumps. Deglaze skillet with heavy cream, whisking to combine. Add chicken stock and cooked chicken thighs, then bring to a simmer. Remove from heat and mix in peas and parsley.
Cover filling in skillet with prepared pie crust, brush crust with egg and place in oven to bake for approximately 25 minutes, or until crust is golden brown.
Serve warm.
