For the crust: Combine dry ingredients in a food processor (or small bowl). Pulse (or cut) in butter until mix resembles coarse crumbs with some pea size pieces. Add ice and blend (or slowly add cold water) until a dough begins to form. Shape into a flat round; cover and refrigerate one to 24 hours.
On lightly floured surface, roll dough into 10- to 12-inch round (same size as your cast iron skillet). Place on parchment paper-lined baking sheet; cover and refrigerate 30 to 60 minutes. Heat oven to 400ºF. Line rimmed baking sheet with foil.
For the filling: Sprinkle chicken with 1 teaspoon of the salt. Heat cast-iron skillet over medium heat until hot. Add 1 tablespoon of the oil; heat until hot. Cook chicken in two batches 3 minutes or until lightly browned on all sides, adding additional 1 tablespoon oil for second batch. (Don’t cook all the way through.) Remove chicken.
Add remaining 2 tablespoons oil. Cook mushrooms and 1 teaspoon of the salt 2 minutes or until mushrooms release juices. Add onion and garlic; cook 1 minute. Stir in Italian seasoning.
Sprinkle with 1/3 cup flour; stir until flour is lightly browned. Add broth and any accumulated juices from chicken, stirring continuously to prevent lumps. Bring to a boil.
Add carrots, potatoes and remaining 1/2 teaspoon salt; reduce heat to medium-low. Add celery. Simmer 6 to 8 minutes or until vegetables are crisp-tender and sauce is of desired consistency. Remove from heat; stir in chicken. Sprinkle peas over filling but don’t stir in (frozen vegetables will help keep dough from melting).
Top filling with cold dough, carefully pressing edges against edge of skillet. Cut several steam vents in center of dough. Place skillet on baking sheet; bake 40 to 45 minutes or until crust is golden brown and filling is bubbly. Let stand 5 minutes before serving.