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1
Wash and dry the portobellos. Cut into 1/2-inch-thick slices. Set aside.
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2
Combine the balsamic vinegar and half of the garlic in a large bowl. Let sit 5 minutes. Stir in 2 tablespoons of the olive oil and the thyme. Add the portobello slices and carefully mix with the garlic mixture. Let sit.
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3
Preheat a grill to medium-high heat.
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4
Place the bell pepper directly on the grill and cook until blackened on all sides, about 12 minutes, turning as needed. Put in a bowl and cover with plastic wrap. Let sit for 10 minutes.
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5
Mix the mayonnaise and the remaining garlic together in a small bowl. Set aside.
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6
Put the rolls on the grill, crust-side up and cook until lightly toasted, 1 to 2 minutes. Transfer to a platter and spread equal amounts of the garlic-mayonnaise over the bottoms.
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7
Whisk together the sherry vinegar and the remaining 2 tablespoons of olive oil in a large bowl. Peel and cut the peppers into 1/4-inch strips. Add the peppers and arugula to the bowl and toss to coat evenly. Season to taste with the salt and pepper. Arrange even amounts over the top of each roll bottom.
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8
Season the mushrooms with salt and pepper and place on the grill and cook, covered, until browned, about 3 minutes. Turn and cook the other side until browned, about 3 minutes. Put one on top of each bread roll. Put the top on and serve with lots of napkins.
