Heat a grill to medium-high (450°). Meanwhile, cut each tofu block crosswise into 6 slices. Lay slices between paper towels, press out excess water, then transfer to a rimmed baking sheet. Whisk sesame oil, soy sauce, brown sugar, cayenne, and garlic powder together in a small bowl and pour over tofu slices, turning to coat both sides.
Whisk dressing ingredients together in a small bowl.
Oil cooking grate, using tongs and a wad of oiled paper towels. Transfer tofu slices to grill and cook until grill marks form on both sides, 2 to 3 minutes per side (tofu may stick a little).
Mix lettuce, cabbage, mushrooms, green onions, and carrot in a large bowl. Toss with three-quarters of dressing. Divide salad among 4 dinner plates and arrange 3 tofu slices, slightly overlapping, on top of each. Drizzle remaining dressing over tofu and sprinkle with sesame seeds.