Prepare a grill for medium-high heat.
For the mushrooms: Pulse the orange juice, vegetable oil, cider vinegar, brown sugar, chipotles and adobo sauce and1/2 teaspoon salt in a mini-food processor or blender until smooth and emulsified.
Put the mushroom caps in 1 or 2 large resealable plastic bags and pour in the marinade. Rub the marinade into the caps, seal the bags and marinate at room temperature while you make the corn salsa, turning the bags over occasionally.
For the corn salsa: Lightly oil the grill grates. Brush the corn evenly with 2 tablespoons vegetable oil and sprinkle with 1/2 teaspoon salt and a few grinds of pepper. Grill, turning occasionally, until the kernels are cooked through and charred in spots, 10 to 15 minutes. Set aside to cool slightly; keep the grill at medium-high heat.
Whisk the mayonnaise, Parmesan and lime zest and juice in a large bowl until combined. Cut off the corn kernels with a paring knife and stir into the mayonnaise mixture. Mix in the cilantro and 1/4 teaspoon salt and set aside.
Remove the mushrooms from the marinade, shaking off any excess, and season on both sides. Pour the marinade into a bowl. Put the mushrooms on the grill cap-side down and grill until very charred, rotating them a few degrees to keep one spot from charring too much if necessary, about 4 minutes. Flip, brush with a little marinade and grill until the bottoms are charred and the mushrooms are tender when pierced with the tip of a paring knife, about 4 more minutes. Transfer the mushrooms to a cutting board, slice thickly and transfer to a platter.
For the taco bar: Serve the mushrooms immediately with the tortillas, corn salsa, Monterey Jack, pico de gallo, radishes, avocados and limes, letting diners build their own tacos.
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