Heat oil in a large oven-proof skillet over medium-high heat. Add onion; cook and stir until soft and translucent, about 5 minutes. Add mushrooms and cook for 1 to 2 minutes. Pour whiskey carefully into the hot skillet. Reduce heat to medium and let whiskey cook down, about 5 minutes.
Whisk sour cream, vegetable broth, and flour together in a bowl. Stir into the skillet. Bring to a boil; reduce heat and simmer until sauce thickens, about 5 minutes.
Combine chicken and green beans in a large bowl; pour in sauce and toss until well coated. Season with salt and pepper. Pour mixture back into the skillet and cover with a tight-fitting lid or aluminum foil.
Bake in the preheated oven until heated through, about 30 minutes. Uncover and scatter onion rings on top. Continue baking until rings are crispy and golden brown, about 15 minutes more. Stir gravy into the hot casserole.