Preheat the griddle. Preheat the fryer. Season the patties with salt and pepper. Place the patties on the hot griddle and cook for 3 to 4 minutes, flip the patties over and continue to cook. Add the onions to the griddle and saute for 4 to 5 minutes or until wilted. Season the onions with salt and pepper. In a mixing bowl, whisk the flour, eggs and beer together. Season the batter with salt and pepper. Season the onions with salt and pepper. Dip each onion ring in the batter, coating completely and letting the excess drip off. Carefully lay the onion rings in the hot oil and fry until golden brown, 3 to 4 minutes, stirring occasionally for over-all browning. Remove the onions rings from the oil and drain on a paper-lined plate. Season the rings with salt and pepper. During the last couple of minutes of cooking, place one piece of cheese on each patty. After the cheese has melted remove the patties from the griddle and set aside. Top each patty with the Sauteed onions. Oil both sides of the bread and place on the griddle. Cook the bread for 1 to 2 minutes on each side or until toasted. Remove the bread from the griddle and place in-between each patty. Serve the patty melts with the onions rings.