Heat the butter and oil in a frypan. Add the onions and cook until soft. Add the diced mushrooms and stir. Add the lemon juice and salt and pepper to taste. Cook until the liquid has evaporated and set aside. Preheat the oven to 200 degrees C.
With a mallet lightly pound the fillets. Sprinkle with salt and pepper on both sides. Heat the 2 tablespoons butter in a frypan and cook until golden on one side, about three minutes, then flip and cook a further two minutes on the other side. Remove from the frypan.
Place the fillets in a baking dish. Cover each with about 2 tablespoons of the mushroom mixture. Top each with a slice of cheese.
To the frypan you used to cook the chicken add the 1 tablespoon onion and the wine, stirring to dissolve any browned bits. Add the cream and simmer 1 minute, then strain over the chicken fillets. Bake in the preheated oven about 5 minutes until cheese is melted.