Melt butter with olive oil in a large skillet over medium-low heat. Add sliced onions and salt, stirring to coat. Cook for 25-30 minutes, stirring every few minutes, until onions are deep golden brown and jammy. If they start to stick, splash in a tablespoon of water. Remove from heat and let cool slightly. Reserve 3 tablespoons for the aioli.

In a small bowl, combine mayonnaise, 3 tablespoons finely chopped caramelized onions, garlic, Dijon, lemon juice, and Worcestershire. Stir until smooth. Taste and adjust with more lemon or salt if needed. Set aside or refrigerate up to 3 days.

Heat the same skillet over medium-high heat. Add ground beef, breaking it into large chunks. Let it sear undisturbed for 3 minutes to develop a crust, then break into smaller pieces and continue cooking until browned and cooked through, about 6-8 minutes total. Season with salt and pepper. Drain excess fat if needed.

Divide lettuce among 4 bowls. Top with warm beef, remaining caramelized onions, tomatoes, pickles, and cheese. Drizzle generously with aioli and serve immediately with extra aioli on the side.