Cut each chicken breast in half vertically so that you have 4 thin chicken breasts. Sprinkle with salt and pepper on both sides.
Once skillet is hot, place on the stove top over medium heat. Add 1/2 tbsp of the oil and sear chicken breasts for 3-4 minutes per side, or until lightly browned. Transfer to a plate.
Turn heat to low and add remaining oil to skillet. Add onions and mushrooms and cook, stirring, until softened, about 5-10 minutes.
Meanwhile, whisk together BBQ sauce, mustard, vinegar, pepper and paprika. Pour over onions and mushrooms and stir to coat. Add chicken back to the skillet and place in the oven to finish cooking, about 10 minutes.
Remove skillet from oven and spoon mushrooms and onions on top of the chicken breasts. Sprinkle evenly with cheese and place back in the oven until cheese is melted.
Serve sprinkled with chopped green onions (optional).