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FOR BALSAMIC PEPPERS AND ONIONS:.
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Prepare a medium-hot fire in a charcoal grill with a cover, or preheat a gas grill to medium-high.
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Heat the oil in a fireproof skillet and add the bell pepper, onion, salt, and pepper.
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Cook, stirring occasionally, until the vegetables are softened, about 15 minutes.
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Add the glace and cook 5 minutes longer.
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Cover the pan with aluminum foil and keep warm.
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FOR THE PATTIES:.
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Combine the garlic, oregano, fennel, paprika, salt, pepper, and cheese in a large bowl.
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Add the wine, pepper sauce, and beef.
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Handling the meat as little as possible to avoid compacting it, mix well.
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Form the mixture into 6 equal oval patties to fit the bread slices, cover, and refrigerate until ready to grill.
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FOR THE AIOLI:.
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Combine all the ingredients in a small bowl and mix well.
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Cover and refrigerate until assembling the patty melts.
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Brush the grill rack with vegetable oil.
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Place the patties on the rack, cover, and cook, turning once, until done to preference, 4 to 5 minutes on each side for medium.
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Butter both sides of the bread slices, place on the rack, and toast one side only until golden brown.
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TO ASSEMBLE:.
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Spread the toasted sides of the bread slices with the aioli.
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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, aioli sides down.
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Return the patty melts to the grill and cook, turning once, until the outside bread is toasted, about 1 minute per side.
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Serve immediately.
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Suggested wine pairing: Zinfandel.
