Barbecue Pulled Mushroom Sandwiches

Name: Barbecue Pulled Mushroom Sandwiches
Ingredients: Vegetable oil, for grill; 2 cups finely shredded green or red cabbage (or a combination); 1 large carrot, peeled and julienned (¾ cup); ¼ cup packed fresh cilantro leaves and thin stems, chopped; 2 tablespoons mayonnaise or Vegenaise; 2 tablespoons apple-cider vinegar; Kosher salt and freshly ground pepper; 1 ½ pounds maitake mushrooms (10 cups), tough stems trimmed but cores left intact, broken into large clusters; ¼ cup extra-virgin olive oil; 1 recipe Speedy Barbecue Sauce, warmed; 4 brioche buns, split