In a bowl, combine the ground beef with the chopped rosemary, salt, and pepper. Divide the mixture into four equal portions and form into patties.
Heat olive oil in a skillet over medium-high heat. Cook the patties on both sides until golden brown and cooked through.
Thinly slice the red onions and sauté them in olive oil until softened. Add the honey and balsamic vinegar, and continue to cook until the onions are caramelized and have a jam-like consistency.
Toast the cut sides of the burger buns until golden brown and crispy.
Place a slice of cheese on top of each patty in the skillet. Cover the skillet and let the cheese melt.
Assemble the burgers: Place iceberg lettuce on the bottom bun, followed by the cheese-topped patty, and then the caramelized red onions. Top with the remaining bun.
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