Brush mushrooms with oil and sprinkle with seasoning blend. To grill mushrooms, place on a hot grill until deep brown, about 10 minutes. Turn and grill until golden brown, another 6-8 minutes.
Meanwhile, combine mayonnaise and ginger in a small bowl. Spread over tortillas. Top with spinach and tomato slices. When mushrooms are done, slice into thin strips and sprinkle with additional seasoning blend if desired. Place mushrooms on wraps and roll up tightly.