Heat 2 tablespoons of butter and the chicken in a large skillet over medium-high heat. Cook until the chicken is cooked through, about 7-10 minutes. Remove from skillet and set aside.
While the chicken cooks, make the stuffing. Bring 2 cups chicken broth and 2 tablespoons of butter in a medium saucepan to a simmer. Remove from heat and mix in stuffing. Set aside.
Preheat the oven to 375°F.
Add remaining 4 tablespoons of butter, onion, celery and carrots to the large skillet over medium heat. Cook until vegetables are tender, about 5-7 minutes. Add garlic, nutmeg, thyme, sage, salt and pepper. Stir to combine and let cook for 1 minute. Sprinkle in flour and mix until combined. Cook for 1 more minute.
While mixing, slowly pour in remaining 2 cups chicken broth, followed by milk. Simmer until thickened.
Remove from heat. Mix in sour cream, green beans, corn and chicken.
Transfer to a 9x13 inch baking dish. Top filling with stuffing and spread evenly.
Cover and bake for 30-35 minutes, or until hot and bubbly around the edges. Remove the cover and bake another 5 minutes.
Serve with gravy, if desired. Enjoy!