Combine mayo, mustard, BBQ sauce, ketchup, and minced jalapeño in a bowl, stirring until smooth and fully incorporated. Set the sauce aside. Thinly slice the two onions into 1/8-inch half-moons and set aside. Divide the ground beef into four equal portions (about 5 ounces each), keeping them loose and not packed—this ensures they'll smash into tender, flavorful patties. Have all ingredients prepped and within arm's reach before you start cooking.
Heat your griddle to medium-high heat. While it preheats, spread the 1/4 cup mayo evenly on both sides of all 8 bread slices. Once the griddle is hot, toast the bread for 2-3 minutes per side until golden and crispy, then transfer to a plate. Reduce heat to medium, melt 3 tablespoons butter on the griddle, then add the sliced onions. Cook for about 20 minutes, stirring occasionally, until deeply caramelized and golden brown. In the final minute, stir in the Worcestershire sauce and granulated sugar to deglaze and add savory depth, then transfer the onions to a plate.
Increase griddle heat to medium-high and melt 2 tablespoons butter. Season your four beef portions generously with kosher salt, black pepper, and garlic powder on both sides. Place them on the hot griddle and let them cook undisturbed for 3 minutes to develop a flavorful crust. Using a sturdy spatula, press each patty down firmly to about 1/4-inch thickness—this creates more surface area for browning and melds the flavors into the beef. I find a quick, confident smash works better than multiple gentle pushes. Flip the patties and cook for another 3 minutes on the second side.
Top each smashed patty with one slice of Swiss cheese followed by one slice of pepper jack cheese. Add a quarter of the caramelized onions from Step 2 on top of each patty. Cook for 1-2 minutes more until the cheese is completely melted and the onions are warm.
Spread a generous spoonful of the special sauce from Step 1 on the inside of each toasted bread slice. Place one loaded patty (with melted cheese and onions) on the bottom slice of each sandwich, then top with the other slice. I like to let these rest for just 30 seconds before cutting—this helps the cheese set and keeps everything from sliding around. Serve immediately while the bread is still warm and the cheese is melted.