Crispy Miso Mushroom Lettuce Wraps

Name: Crispy Miso Mushroom Lettuce Wraps
Ingredients: 4 tablespoons extra-virgin olive oil; 2 pounds assorted mushrooms, finely diced; Kosher salt and freshly ground black pepper; 1/2 stick (4 tablespoons) unsalted butter, at room temperature; 2 tablespoons white miso; 1 teaspoon light brown sugar; 2 tablespoons rice wine vinegar; One 8-ounce can water chestnuts, drained and finely chopped; 1 large carrot; 1 large radish, such as watermelon radish; 1 green apple, cored; 1 tablespoon fresh lime or lemon juice, plus 1 lime, quartered for serving; 2 heads lettuce, such as butter lettuce, baby romaine or Little Gem; 1 bunch mint; 1 bunch cilantro; Sriracha or plum sauce, for serving