Combine the ketchup, brown sugar, balsamic vinegar, Worcestershire and paprika in a small saucepan. Cook over medium heat, stirring occasionally, until the sugar is completely dissolved, about 10 minutes. Set the BBQ sauce aside.
Meanwhile, heat 2 tablespoons of the canola oil in a large skillet over medium-high heat. Add the mushrooms and cook, stirring occasionally, until browned all the way through, 7 to 10 minutes. Set aside.
In the same skillet, add the remaining 1 tablespoon oil and the garlic and onion; cook, stirring occasionally, until the onions have softened and browned slightly, about 5 minutes. Return the mushrooms to the skillet; add the BBQ sauce and stir.
Place the bottom buns on a plate and top with the BBQ mushrooms and iceberg lettuce. Close the sandwiches and serve.