Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.
To make the peppers and onions, heat the oil in a fireproof skillet. Add the bell pepper, onion, salt, and pepper. Cook, stirring occasionally, until the vegetables are softened, about 15 minutes. Add the glace and cook 5 minutes longer. Cover the pan with aluminum foil and keep warm.
To make the patties, combine the garlic, oregano, fennel, paprika, salt, pepper, and cheese in a large bowl. Add the wine, pepper sauce, and beef. Handling the meat as little as possible to avoid compacting it, mix well. Form the mixture into 6 equal oval patties to fit the bread slices, cover, and refrigerate until ready to grill.
To make the aïoli, combine all the ingredients in a small bowl and mix well. Cover and refrigerate until assembling the patty melts.
Brush the grill rack with vegetable oil. Place the patties on the rack, cover, and cook, turning once, until done to preference, 4 to 5 minutes on each side for medium. Butter both sides of the bread slices, place on the rack, and toast one side only until golden brown.
To assemble the patty melts, spread the toasted sides of the bread slices with the aïoli. Top 6 of the slices with equal portions of the peppers and onions, the patties, a few basil leaves, an equal portion of the cheese, and the remaining bread slices, aïoli sides down. Return the patty melts to the grill and cook, turning once, until the outside bread is toasted, about 1 minute per side. Serve immediately.
Suggested wine pairing: Zinfandel