Preheat the oven to 400°F.
Combine the sesame oil, garlic, ginger, salt, and smoked paprika. In a large baking dish, thoroughly rub this mixture on all of the mushrooms, making sure they are evenly coated. Be sure to get the spice mixture between the leaves of the mushrooms as well. Spread out in the baking dish, and roast for about 20 minutes. While the mushrooms roast, I like to occasionally spoon some of the juices that are released while they cook over the mushrooms to keep them evenly seasoned.
Prepare Roasted Tomato Balsamic BBQ Sauce: In a medium sized baking pan, combine all of the ingredients. Roast in the oven for about an hour, stirring occasionally until the tomatoes have completely broken down, and a lot of the liquid has evaporated.
Add Roasted Tomato Balsamic BBQ Sauce to a blender and process until smooth. Adjust salt, and serve with the oyster mushrooms.