In a large skillet, heat the oil. Add the sliced onion and a pinch of salt, cover and cook, stirring a few times, until the onion is crisp-tender and just turning translucent, 5 minutes. Stir in the hot sauce and beef broth and remove from the heat. Season with salt.
Heat a large griddle. Spread the butter on 1 side of each bread slice and griddle buttered side down over moderate heat until toasted, about 2 minutes. Transfer the toasts to a plate. Add the bacon to the griddle and cook over moderate heat until crisp, about 4 minutes per side; drain on paper towels. Wipe off the griddle.
Season the beef patties with salt and cook on the griddle over moderately high heat until medium-rare, about 2 minutes per side. Top each burger with 1/4 cup of smoked Gouda and cook until the cheese melts, 1 minute.
Set the burgers on the unbuttered sides of 4 toasts. Top with the bacon, onion and Sweet Barbecue Sauce. Close the burgers and serve, passing more barbecue sauce at the table.