1. To prepare tofu, place tofu slices on several layers of paper towels; cover with additional paper towels. Top with a cutting board; place a heavy skillet on top of cutting board. Let stand 45 minutes, pressing down occasionally. Cut each tofu slice into 10 cubes; arrange in a single layer in a shallow dish.
2. Preheat oven to 375°.
3. Combine brown sugar and next 4 ingredients (through garlic) in a small bowl. Pour sugar mixture over tofu in dish, and turn to coat. Let stand for 25 minutes. Arrange tofu in a single layer on a baking sheet coated with cooking spray. Bake at 375° for 35 minutes, turning after 15 minutes. Cool completely.
4. To prepare salad, heat 1 teaspoon sesame oil in a heavy saucepan over medium-high heat. Add barley to pan; cook 3 minutes or until lightly toasted, stirring frequently. Add 5 cups water and 1/2 teaspoon salt; bring to a boil. Reduce heat, and simmer 45 minutes or until barley is tender. Drain; cool completely. Place barley in a large bowl.
5. Cook green beans in boiling water for 4 minutes or until crisp-tender. Drain and plunge green beans into ice water. Drain well; pat dry. Add beans to barley.
6. Heat a grill pan over medium-high heat. Brush tops of mushrooms with 2 teaspoons oil. Add mushrooms to pan, oiled side down. Cook 5 minutes or until browned. Sprinkle evenly with remaining 1/4 teaspoon salt. Cool slightly; slice mushrooms thinly. Add tofu, mushrooms, and green onions to barley mixture.
7. Combine remaining 1 tablespoon oil, rice wine vinegar, 2 tablespoons soy sauce, agave nectar or honey, grated ginger, and minced garlic in a small bowl, stirring with a whisk. Drizzle over barley mixture in bowl; stir well to combine. Serve at room temperature or chilled.