In a bowl, combine the ground beef with salt, black pepper, oregano, and one tablespoon of olive oil. Shape the mixture into 4 equal-sized patties.
In a hot skillet or on a grill, cook the patties for 3-4 minutes on one side, then flip.
After flipping, place a slice of mozzarella cheese on each patty. Cover the skillet to allow the cheese to melt.
Halve the hamburger buns and spread butter on each half. Toast them in a skillet until golden brown.
Spread pesto on the bottom half of each bun, then top with a handful of arugula.
Place the mozzarella-topped patty on the arugula, then add tomato slices and thinly sliced red onion.
Drizzle the top half of the bun with a bit of balsamic vinegar, then place it on top of the burger.
Serve immediately with French fries or Italian Caprese salad.
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