Make the Caramelized Onions

In a large skillet, heat olive oil over medium heat.

Add both types of onions and stir to coat in the oil. Cook, stirring occasionally, for about 12–15 minutes until they start to soften and turn golden.

Sprinkle the salt and sugar over the onions, reduce the heat to medium-low, and continue to cook for another 25–30 minutes, stirring occasionally, until the onions become deeply golden and jam-like.

Pour in the vegetable stock, let it simmer until absorbed, and the onions become rich and glossy.

Remove from heat and set aside.

Prepare the Garlic Sauce

In a small bowl, whisk together the mayonnaise, olive oil, and minced garlic until smooth.

Cover and refrigerate until ready to use.

Cook the Burger Patties

Shape the ground beef into 6 equal patties.

Season both sides generously with salt and black pepper.

Heat vegetable oil in a large skillet or cast-iron pan over medium-high heat.

Cook the patties for about 3–4 minutes per side, until browned and crisp on the outside.

Flip, then top each patty with shredded cheese and continue cooking for 2–3 more minutes, until the cheese melts

And the burgers reach your preferred doneness.

Transfer the burgers to a plate and cover to keep warm.

Toast the Buns

Add butter to the same pan and let it melt.

Place the buns cut-side down and toast for 1–2 minutes, until golden brown and crisp.

Assemble the Burgers

Place a burger patty on each bottom bun.

Top with a generous spoonful of caramelized onions.

Add a handful of fresh arugula.

Spread the garlic sauce on the top bun and close the burger.

Serve immediately while hot.