To prepare tofu, combine the first 4 ingredients, stirring with a whisk. Cut tofu lengthwise into 4 equal slices. Carefully place tofu in an 11 x 7-inch baking dish. Pour the soy sauce mixture over tofu. Cover and refrigerate 30 minutes, carefully turning once.
Remove tofu from dish; discard marinade. Place tofu on a baking sheet coated with cooking spray. Broil 10 minutes on each side or until tofu is lightly browned.
To prepare greens, heat oil in a large nonstick skillet over medium heat. Add mushrooms, ginger, pepper, and garlic; cook 1 minute, stirring frequently. Add chard and sesame seeds; cook 3 minutes or until chard wilts, stirring frequently. Stir in 2 teaspoons soy sauce.
Place 1/4 cup chard mixture on each of 2 plates; top each serving with 1 tofu slice. Repeat layers with remaining chard and tofu.
I cut tofu horizontally, which made it easier to handle and marinated for two hours. Doubled the ginger and used chinese mustard greens (tender and peppery) instead of chard, and followed others' advice to save the marinade to drizzle on top of everything. It was okay, and we enjoyed it, but it's nothing to rave about. Served with short-grain brown rice and some melon slices.