Melt butter in sauce pan over medium to medium high heat and add onions and apples, cinnamon, bay. Salt and pepper, a light sprinkle of nutmeg, sugar and lemon juice, cook to sauce-like consistency stirring occasionally 20-25 minutes, douse with liquor.
Season pork with salt and pepper and coat with flour, egg beaten with milk and breadcrumbs seasoned with nutmeg and parsley. Shallow fry the cutlets in thin layer of vegetable oil until crisp and golden. Squeeze lemon juice over top of cutlets just before you're ready to build sandwiches.
Lightly toast the rolls under a broiler then divide the cutlets over top of the toasted roll. Spoon some apple onion mixture over top the cutlets then top with shredded cheese and broil until cheese is melted.