Preheat oven to 425°F.
Toss pepper and onion slices with olive oil, salt and pepper and spread out on two foil-lined baking sheets. Roast until soft and charred in spots, about 20 minutes, stirring once or twice. Set aside to cool slightly.
Arrange 8 slices of bread on a board and spread each with butter. Flip all of the slices and spread with mayonnaise. Divide avocado across half the of slices. Top with peppers and onions. Pile 1/3 cup shredded cheese evenly on top of the peppers. Top with remaining bread slices, mayonnaise side down.
Preheat a large skillet over medium heat. Carefully transfer one sandwich to the skillet and cook until bread is golden and cheese is starting to melt, 2-3 minutes. Flip and cook another 2-3 minutes, until cheese is fully melted. Repeat with the remaining 3 sandwiches.
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