Make the sauce

• In a medium bowl, combine the mayonnaise, mustard, ketchup, caramelized onions, pickles, salt, pepper, sugar and vinegar.
• Cover and refrigerate while you prepare the rest.

Caramelize the onions

• Heat a small pan over low heat with the oil and butter.
• Cook the onions slowly, stirring occasionally, until soft, golden brown and sweet.
• Set aside.

Shape the patties

• Divide the ground beef into 4 equal portions.
• Roll each into a ball, then flatten into thin croissant-shaped patties about 1/4-inch thick, wider than the bun as they will shrink.
• Season one side generously with salt and pepper and keep chilled until ready to cook.

Toast the croissants

• Cut the croissants in half.
• Lightly butter the cut sides.
• Toast them cut-side down in a dry skillet or griddle over medium heat until golden and crispy, then set aside.

Cook the patties

• Heat a skillet or griddle over medium-high heat and add a thin layer of oil.
• Add the patties, two at a time, without overcrowding.
• Cook on the first side until browned, then spread mustard evenly over them.
• Flip and immediately add a slice of cheese to each patty.
• Cook until the cheese is melted and the patty is cooked through.

Assemble the burgers

• Spread the sauce on the toasted croissants.
• Add the patties with melted cheese, caramelized onions, lettuce and tomato.
• Brush the croissants with garlic butter and enjoy.