Toss the onions and peppers with the olive oil and salt and pepper to taste.
Place the onions and peppers in a grill pan or other implement to keep them from falling through the grill. Place on the grill.
Stir the onions occasionally, and grill for 10 to 12 minutes, until they are browned and softened. Remove from grill and set the onion mixture aside in a bowl.
When the onions are on the grill, shape the burgers into two patties. Add a bit of salt to taste.
Place the burgers on the grill. Grill covered for about 6 minutes, then turn the burgers to the other side. Grill for about 6 more minutes. When the burger is just about cooked, add the cheese and grill covered for another minute or so until the cheese has begun to melt.
Remove the burgers from the grill. Serve the burger on a bun topped with a portion of the grilled onions. Pass the ketchup at the table and enjoy with a bold red wine.