To make the Secret Sauce, stir together the mustard, mayonnaise, barbecue sauce, and hot sauce in a small bowl. Cover and refrigerate for up to 3 days.
In a large bowl, mix together the ground beef, Worcestershire sauce, salt, and pepper. Shape the mixture into 6 oval patties.
In a medium cast-iron skillet over medium heat, melt 2 tablespoons of butter. Add the thinly sliced onions and cook, stirring occasionally, for approximately 35 minutes or until the onions are soft and golden brown.
In a large cast-iron skillet over medium-high heat, cook the patties until browned and cooked through, approximately 2 minutes per side. Remove the skillet from the heat and remove the patties from the skillet. Wipe out the skillet.
Layer 1 slice of bread with 1 tablespoon of Secret Sauce, 2 to 3 tablespoons of caramelized onions, 1 slice of cheese, 1 patty, and another 1 tablespoon of Secret Sauce. Top with another slice of bread. Repeat with the remaining bread, Secret Sauce, caramelized onions, cheese, and patties.
Heat the skillet over medium-high heat and melt 2 tablespoons of butter in the skillet. Working in batches, cook the sandwiches, flipping once, until golden brown and heated through, approximately 3 minutes per side. Add the remaining butter to the skillet as needed.