Mix mayonnaise, ketchup, mustard, caramelized onions, pickles, vinegar, sugar, salt, and pepper. Cover and refrigerate.
Heat a pan over low heat with olive oil and butter. Add onions, sugar, and salt. Cook 15 to 20 minutes, stirring occasionally, until soft and golden. Set aside.
Divide ground beef into 4 portions. Roll into balls and flatten into thin patties wider than the croissants. Season generously with salt and pepper. Keep chilled until ready to cook.
Slice croissants in half. Brush cut sides with garlic butter. Toast cut side down until golden. Remove and set aside.
Heat skillet or griddle over medium high heat with a thin layer of oil. Cook patties for 2 minutes until browned. Spread mustard on cooked side, flip, add cheese, and cook 1 to 2 minutes until melted.
Assemble with sauce, patties, lettuce, tomatoes, caramelized onions, and croissants. Serve hot.
