In a resealable plastic bag, combine the shiitakes, soy, vinegar, oil, brown sugar, hot red pepper flakes, and garlic. Seal the bag and marinate for at least 30 minutes at room temperature and up to overnight in the refrigerator.

Transfer the mushrooms and marinade to a small skillet over medium-high heat. Cook the mushrooms, mixing often, until the liquid from the marinade and the mushrooms evaporates and the mushrooms darken in color, about 7 minutes. Remove from the heat and set aside to cool.

In a large mixing bowl, whisk together the dressing ingredients. Add the thinly sliced Brussels sprouts, followed by the cooked mushrooms, and toss well. Season with salt and pepper to taste (you shouldn’t need much salt due to the soy sauce in the mushrooms). Arrange the salad on plates and garnish each serving with a couple good pinches of pecorino and more black pepper.