Heat oven to 350°F. Set a 9x13-inch baking dish nearby.
Pat chicken dry. Season all over with salt, pepper, garlic powder, and thyme.
Arrange chicken in the dish in a single layer. Cover evenly with Swiss cheese.
Whisk soup, sour cream, broth, and Dijon in a bowl until smooth.
Pour the sauce over the chicken and cheese, spreading to coat.
Toss stuffing with melted butter until moistened. Sprinkle evenly over the casserole.
Bake 35–45 minutes, until topping is golden and chicken reaches 165°F. Rest 5 minutes.
Scatter parsley over the top and serve warm.