Heat a skillet over medium heat with olive oil, 2 turns of the pan. Add onions and season with salt and a little pepper. Cook onions until caramel in color and very sweet, 35-40 minutes.
Heat a cast-iron skillet or griddle over medium-high heat. Loosely form 4 patties of beef and place them onto a hot grill, flattening them evenly with a spatula. Season with Kosher salt and pepper, top with caramelized onions and press onions into the burger. Cook to crust and brown the meat, 3 minutes. Flip, season and press and crisp the opposite side; cook 2-3 minutes more.