Heat the oven to 400°F, then warm the olive oil over medium-high heat in a large cast-iron or oven-safe skillet. Season the chicken with paprika, garlic powder, onion powder, oregano, thyme, 1/2 teaspoon salt, and 1/4 teaspoon pepper, then cook it for 5 to 6 minutes per side until golden and cooked through; transfer it to a cutting board, cool slightly, and shred or chop it. Look for a browned crust on both sides of the chicken before removing it from the skillet.
Lower the heat to medium and melt 3 tablespoons butter in the same skillet, then cook the onion, celery, and carrot for 5 to 7 minutes. Stir in the garlic, crushed red pepper flakes, poultry seasoning, and remaining salt and pepper for 30 seconds to 1 minute, then work in the flour and cook for 1 to 2 minutes. The vegetables should be softened, and the flour should smell toasted and look lightly golden with no dry patches.
Whisk in the chicken stock slowly until the skillet is smooth, bring it to a simmer, then stir in the heavy whipping cream, chicken, and peas. Simmer for 1 to 2 minutes, then take the skillet off the heat. The gravy should gently coat the back of a spoon and wobble slightly when the skillet is nudged.
Stir the biscuit mix, sharp cheddar cheese, and buttermilk in a large mixing bowl, leaving the seasoning packet aside. Drop heaping spoonfuls of dough across the hot filling, spacing them over the surface rather than pressing them down. The dough should sit on top in distinct mounds, with bubbling gravy visible between the biscuits.
Bake the skillet for 20 to 25 minutes until the biscuit topping is fully baked and golden. While it bakes, melt the remaining 3 tablespoons butter in a small microwave-safe bowl and stir in the garlic herb seasoning packet. The biscuit tops should rise high and turn a deep, even golden brown.
Brush the hot biscuits immediately with the seasoned butter, scatter the parsley over the top, and let the pot pie stand for 10 minutes before serving. The butter will sizzle softly as it hits the crust. The filling will settle during the rest, making it thick and easier to spoon into bowls.