Preheat oven to 350°F. Grease a 9×13-inch baking dish.
Melt the butter. Use 1 tbsp to sauté onion and celery in a skillet until tender, 5 minutes. Reserve remaining melted butter.
Whisk condensed soups, sour cream, 3/4 cup broth, garlic powder, sage, thyme, salt, and pepper in a large bowl.
Fold in the shredded chicken and the cooked vegetables until evenly coated.
Combine stuffing mix with the reserved melted butter and the remaining 1/2 cup broth. Toss until lightly moistened and fluffy.
Spread half of the stuffing mixture in the prepared dish.
Spoon the chicken mixture evenly over the stuffing layer.
Top with the remaining stuffing. Press gently to level.
Bake 35–40 minutes until bubbling at the edges and lightly browned. Rest 10 minutes, then garnish with parsley and serve.