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.