Drizzle butter into a casserole dish. Season chicken with salt, pepper, and 1 tablespoon garlic. Wrap chicken thighs in prosciutto, and place in casserole dish.
Sprinkle mushrooms and remaining garlic on top of chicken.
Bake in a preheated oven until juices run clear, about 1 hour.
Remove chicken to a platter, and cover with aluminum foil to keep warm.
Pour drippings from casserole into a skillet set over medium-low heat.
Whisk in wine and sour cream, and cook until warmed through, about 5 to 7 minutes.