Prepare the mushrooms by wiping them down with a wet paper towel. Pop off the mushroom top and put it to the side for another use. Pull off the thicker fiberouse base of the mushroom and discard. Holding the mushroom with a hand on either side, use your thumbs to press into the center of the mushroom and split it in half. Now there should be clear raw edges to the mushroom that you can peel and pull apart like you would shred chicken. Repeat these steps over a large bowl with the mushrooms until they are all shredded.
In a small bowl combine the soy sauce, apple cider vinegar, grated garlic, two teaspoons of the smoked paprika, and the oil. Whisk to combine, pour over the shredded mushrooms and toss thoroughly to coat all of the mushrooms.
Cover the bowl of mushrooms and place in the fridge for 20 minutes to marinade. Preheat the oven to 400° F.
Prepare a large sheet pan with parchment paper. Pour the marinated mushrooms on the parchment paper and spread into an even layer. Bake the mushrooms for 20-25 minutes until tender.
Preheat the broiler and toss the baked mushrooms with the BBQ sauce and spread out evenly again. Broil until the mushrooms are caramelized and crispy on the edges, 3 to 5 minutes. Return the oven to 400° F.