Preheat oven to 425°F. Heat 2 tablespoons butter and 1 tablespoon oil in a 12-inch cast iron skillet over medium heat; add diced onion, carrots, and celery, season lightly with salt, and cook until softened, about 6–8 minutes.

Add minced garlic and cook 30 seconds; push vegetables aside, add 3 tablespoons butter and sprinkle 3 tablespoons flour over the pan, stirring constantly for 1–2 minutes to form a pale roux.

Slowly whisk in 2 cups low-sodium chicken stock until smooth, then stir in Dijon, thyme, and bring to a gentle simmer.

Stir in 1 to 1 1/2 pounds chopped cooked chicken and 1 cup frozen peas; simmer 4–6 minutes to combine flavors.

Mix 1 tablespoon cornstarch with 2 tablespoons cold water and stir into the simmering filling; cook 1–2 minutes until thickened, then stir in 1 teaspoon lemon juice and adjust salt and pepper.

After sautéing vegetables, push them aside, add 1 tablespoon oil, and sear 1.5 pounds cut chicken thighs 2 minutes per side; then proceed with the roux and simmer until chicken reaches 165°F.

In a bowl combine 2 cups flour, 1 tablespoon baking powder, 1 teaspoon kosher salt, and 2 tablespoons sugar; cut in 6 tablespoons cold butter until pea-sized crumbs form.

Stir in 3/4 cup cold buttermilk until dough just comes together; turn onto a floured surface, fold twice to create layers, and pat to a 9-inch circle about 1/2-inch thick; cut into 6–8 biscuits.

Brush biscuits with beaten egg or milk if desired and arrange on top of the filling leaving small vents between pieces (or place one round top and crimp edges).

Bake in the preheated 425°F oven for 18–22 minutes until biscuit tops are golden and filling bubbles at the edges; tent with foil if biscuits brown too quickly.

Remove and let rest 8–10 minutes before serving. Garnish with chopped parsley if desired.