For the onion-pickle marmalade: Melt the butter in a medium saucepan over high heat. Add the onions and pickles and cook, stirring occasionally, until the onions are translucent, about 5 minutes. Add the vinegar and cook until the mixture has reduced and become sticky, about 10 minutes. Remove from the heat and set aside.
For the burgers: Mix the butter with the beef, salt and pepper in a medium bowl until thoroughly combined. Form 6 equal patties with your hands.
Heat a griddle over medium-high heat until hot. Add the patties and cook until browned on one side, 3 to 4 minutes. Flip and cook the other side until caramelized, 3 to 4 minutes. Flip the patties again. Top 1 patty with a 4-ounce mound of the cheese and cover with a small metal bowl until the cheese melts and cascades around the patty to form a cheese "skirt." When the cheese skirt is crispy around the edges, carefully remove the bowl and transfer the patty with the attached cheese skirt to a plate. Repeat with the remaining patties and cheese.
For the sauce: Stir the mayonnaise and steak sauce in a small bowl until combined. Dollop an equal amount on the bottom buns. Top with the lettuce, tomatoes and cheese-skirt patties. Dollop with the marmalade, close the burgers and serve right away.