Wipe the mushroom caps clean and use a small spoon to scrape out the gills on the underside. Place 3 tablespoons of the vinegar in a small bowl and whisk in the minced garlic, thyme, and ¼ cup olive oil. Divide the mixture between 2 large plastic bags with resealable closures. Put 2 mushroom caps in each bag and seal, pressing out the excess air. Marinate the mushrooms at room temperature for at least 1 hour and up to 2 hours, turning the bags often.
Prepare a charcoal fire and let the coals burn down to a gray ash. Lightly oil the grill. Set the peppers on top and grill, turning often, until the skins are blackened on all sides, 10 to 15 minutes. (Alternatively, preheat the broiler to high and place the broiling pan 4 to 5 inches from the source of heat. Set the peppers on the pan and broil, turning often, until blackened on all sides, 12 to 15 minutes.) Place the charred peppers in a large bowl and let stand, covered with plastic wrap, until cool enough to handle. Peel, core, and seed the peppers and then cut into thin strips. Toss the strips with salt and pepper.
Remove the mushrooms from the marinade, pat dry, and season with salt and pepper. Place the mushrooms cap sides up on the grill and cook until dark and slightly softened, 2 to 3 minutes. Turn and grill the other side until tender, 2 to 3 minutes longer. (Alternatively, the mushrooms can be grilled in a hot, well-seasoned ridged grill pan over medium-high heat for about 5 minutes per side, but grill the bread first; see below.) When the mushrooms are almost tender, sprinkle a quarter of the cheese into the cavity of each cap.
While the mushrooms are grilling, brush the bread with olive oil on both sides. Grill until lightly toasted, turning often, 3 to 5 minutes. Rub one side of each slice with the cut garlic clove while hot. (If using a grill pan, grill the bread first and then the mushrooms.)
Add the reserved tablespoon of vinegar to the mayonnaise. Stir well and season with salt and pepper. Spread on the garlic-rubbed side of each slice of bread. Top half the bread slices with a mushroom cap, equal amounts of the peppers, and basil. Place the other bread slices on top. Cut the “burgers” in half and serve.