Heat your grill or a grill pan to high and prepare the mushrooms. In a large bowl, whisk together the oil, soy sauce, vinegar, honey, Dijon, cumin, and smoked paprika. Season with pepper and toss in the mushrooms to coat. Set aside to marinate while you prepare the slaw.
For the slaw, whisk together the mayonnaise, vinegar, sugar, an onion. Toss in the cabbage and carrot and season to taste with salt and pepper. Set aside.
Lightly oil the grates of your grill and transfer on the mushrooms. Press down with a spatula and cook the mushrooms for about 2 minutes per side or until well charred and tender. Remove the mushrooms to a bowl and, using two forks, pull the mushrooms into bite sized pieces. Pour over some or all the barbeque sauce and toss to combine.
To build the sandwiches, divide the mushrooms over the buns and top with the slaw.