Form the beef into 4 oblong patties (sized to fit the bread slices); season with salt, pepper, and Italian Seasonings. Fry or broil to desired degree of doneness, about 5 minutes per side for medium-well.
Place a patty on each of 4 slices of bread; divide the onions over each patty, then top with cheese slices. Close the sandwich. Lightly brush butter on the outside of the bread; sprinkle both sides with the Parmesan cheese.
Heat a skillet over medium heat; lightly brush with olive oil; place the sandwich into the skillet and cook until nicely browned; flip and repeat with the other side.
*To caramelize the onions, heat 1 tablespoon extra virgin olive oil and 1 tablespoon butter together in a large sauté pan over medium heat. When melted, add the onions and stir well. Once the onions start to sizzle, reduce the heat to medium-low and slowly cook the onions to a deep brown, stirring occasionally (about 15 - 20 minutes). Keep the heat fairly low to keep the onions from burning. The slow cooking helps them develop a deep sweet flavor.