Bring 3 cups water to a boil in a medium, heavy saucepan. Add barley; reduce heat, and simmer 30 minutes or until barley is tender. Drain and cool slightly.
Combine vinegar, 1 tablespoon brown sugar, 1 tablespoon soy sauce, 1 tablespoon oil, 1 teaspoon ginger, and 1 garlic clove in a large bowl, stirring well with a whisk. Add chard and barley; toss well to coat. Cover and chill 1 hour.
Preheat oven to 375°.
Cut tofu crosswise into 5 (1-inch-thick) slices. Place tofu slices on several layers of paper towels; cover with additional paper towels. Let stand 20 minutes, pressing down occasionally.
Cut each tofu slice into 1/2-inch cubes. Combine remaining 1 teaspoon sesame oil, remaining 2 tablespoons brown sugar, remaining 2 tablespoons soy sauce, remaining 1 teaspoon ginger, remaining garlic clove, and pepper in a medium bowl, stirring well with a whisk. Add tofu; toss to combine. Let stand 10 minutes.
Arrange tofu in a single layer on a foil-lined baking sheet coated with cooking spray. Bake at 375° for 30 minutes or until tofu is browned on all sides, stirring three times.
Divide barley mixture evenly among 4 plates, and top evenly with tofu. Sprinkle evenly with sesame seeds and onions.