Heat a large skillet over medium heat and add two tablespoons of olive oil. Once hot, add thinly sliced yellow and red onions, cooking for 12–15 minutes until soft and starting to brown. Sprinkle a pinch of salt and a teaspoon of sugar over the onions, then reduce the heat to low. Continue cooking for 25–30 minutes, stirring occasionally, until the onions are deeply caramelized and have a jam-like consistency.

In a medium bowl, combine half a cup of mayonnaise, one tablespoon of olive oil, and three cloves of minced garlic. Mix well until smooth and creamy. Cover the bowl with plastic wrap and refrigerate for at least 30 minutes.

While the garlic sauce chills, shape the ground beef into six equally sized patties, about one inch thick. Season both sides with salt and black pepper. Heat two tablespoons of vegetable oil in a heavy skillet over medium-high heat. Cook the patties for 3–4 minutes on each side for medium-rare, adding slices of cheese on top during the last minute to melt beautifully.

In the same skillet, add one tablespoon of unsalted butter and let it melt over medium heat. Split the burger buns in half and place them cut side down in the skillet. Toast them for 2–3 minutes until golden brown.

To assemble your gourmet creation, place a cooked patty on the bottom half of each toasted bun. Generously top with a heap of the sweet caramelized onions and a handful of fresh arugula for a peppery kick. Finally, spread a dollop of the creamy garlic sauce on the top half of the bun before placing it gently on top.