Heat 2 tablespoons of the olive oil in a large skillet over medium-high heat, swirling to coat, until the oil is about to smoke. Add half of the mushrooms in a single layer and let them cook, undisturbed, for 2 minutes. Continue to cook, stirring occasionally, until crispy, about 8 minutes. Season with salt and pepper.
Meanwhile, combine the butter with the miso and brown sugar in a small bowl and mash until well mixed.
Add half of the miso butter to the skillet, toss to coat the mushrooms and cook until the butter is bubbling and starting to brown and the mushrooms are glistening, 1 to 2 minutes. Add 1 tablespoon vinegar and scrape up the browned bits from the bottom of the skillet. Add half of the water chestnuts and cook, stirring, until warmed through, 1 to 2 minutes. Transfer to a serving bowl and keep warm.
Wipe out the skillet and repeat the cooking process with the remaining 2 tablespoons olive oil, mushrooms, miso butter, 1 tablespoon vinegar and water chestnuts. Transfer to the serving bowl and keep warm.
Slice the carrot into 2- to 3-inch lengths. Slice off the rounded sides from the carrots and discard. Cut the carrots into 1/8-inch-thick planks, lay them flat and slice into thin matchsticks. Slice off and discard the rounded sides from the radish and apple then cut into matchsticks. Toss the apples with the lime or lemon juice in a small bowl.
Place the bowl of crispy mushrooms on a cutting board or tray. Arrange the lettuce, carrot, radish, apple, mint, cilantro and limes around the bowl. Serve with Sriracha or plum sauce and let guests make their own lettuce wraps at the table.