Add 1 tbsp of oil to a cast-iron skillet and place it over high heat.
Add the chicken and 1 teaspoon of salt. Stir occasionally while cooking the chicken until it’s browned and no longer pink inside. The internal temperature of the chicken should be 165°F for white meat and 175-180°F for dark meat.
Remove the chicken from the skillet and set it aside.
Reduce the heat to medium, then immediately add the chopped onion, mushrooms (if using), carrots, celery, garlic, the remaining 1 ½ teaspoons of salt, and black pepper. Stir to combine.
Cook until all the vegetables have softened, about 5 minutes.
Add the butter for the filling. Once it's melted, stir in the flour and cook for a minute.
Add the broth/stock and milk. Bring it to a simmer and gently cook until the filling has thickened. Remove from the heat
Add the frozen peas (you don't need to thaw them), dried thyme, dried tarragon, and the chicken. Cover and set aside while you work on the dough