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1
Wipe the mushroom caps clean and use a small spoon to scrape out the gills on the underside.
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2
Place 3 tablespoons of the vinegar in a small bowl and whisk in the minced garlic, thyme, and 1/4 cup olive oil.
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3
Divide the mixture between 2 large plastic bags with resealable closures.
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4
Put 2 mushroom caps in each bag and seal, pressing out the excess air.
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5
Marinate the mushrooms at room temperature for at least 1 hour and up to 2 hours, turning the bags often.
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6
Prepare a charcoal fire and let the coals burn down to a gray ash.
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7
Lightly oil the grill.
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8
Set the peppers on top and grill, turning often, until the skins are blackened on all sides, 10 to 15 minutes.
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9
(Alternatively, preheat the broiler to high and place the broiling pan 4 to 5 inches from the source of heat.
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10
Set the peppers on the pan and broil, turning often, until blackened on all sides, 12 to 15 minutes.)
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11
Place the charred peppers in a large bowl and let stand, covered with plastic wrap, until cool enough to handle.
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12
Peel, core, and seed the peppers and then cut into thin strips.
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13
Toss the strips with salt and pepper.
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14
Remove the mushrooms from the marinade, pat dry, and season with salt and pepper.
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15
Place the mushrooms cap sides up on the grill and cook until dark and slightly softened, 2 to 3 minutes.
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16
Turn and grill the other side until tender, 2 to 3 minutes longer.
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17
(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.)
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18
When the mushrooms are almost tender, sprinkle a quarter of the cheese into the cavity of each cap.
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19
While the mushrooms are grilling, brush the bread with olive oil on both sides.
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20
Grill until lightly toasted, turning often, 3 to 5 minutes.
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21
Rub one side of each slice with the cut garlic clove while hot.
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22
(If using a grill pan, grill the bread first and then the mushrooms.)
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23
Add the reserved tablespoon of vinegar to the mayonnaise.
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24
Stir well and season with salt and pepper.
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25
Spread on the garlic-rubbed side of each slice of bread.
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26
Top half the bread slices with a mushroom cap, equal amounts of the peppers, and basil.
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27
Place the other bread slices on top.
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28
Cut the burgers in half and serve.