Preheat oven to 400 degrees.
Heat 1 tablespoon butter and 1 tablespoon oil in a 10-inch cast-iron (or other ovenproof) skillet over medium-high heat.
Season the chicken with salt and black pepper and place in the preheated skillet.
Cook, stirring often until the chicken is nicely browned and cooked through. Transfer to a plate.
Add 1 tablespoon butter and the remaining 1 tablespoon oil to the skillet. Add the onion, reduce heat to medium-low and cook, 4-5 minutes or until softened and translucent.
Add the cubed potato and saute 3-4 minutes.
Add the remaining 2 tablespoons butter and allow to melt.
Sprinkle the flour over the potatoes and onions and cook 1-2 minutes, stirring often.
Add the chicken broth. Bring it to a simmer. Cook 4-5 minutes or until thickened.
Add the mixed vegetables, cream and chicken back to the skillet. Heat through.
Add shredded Cheddar cheese by the handful, stirring until cheese is melted. (Sauce will continue to thicken with the addition of the cheese.)
Stir in dry mustard and parsley.
Remove from heat.
Place pie crust over the top. Crimp the edges as desired.
Brush with egg wash.
Bake 25-30 minutes or until pastry is golden and pie is visibly bubbling.