1. Bring a large pot of water to a boil. Add the barley and 1/2 teaspoon of the salt; boil for 30 minutes. Add the farro; boil for an additional 20 to 25 minutes or until both grains are just tender. Drain.
2. Meanwhile, bring 6 tablespoons of water to a boil in a small saucepan; add the bulgur. Bring the liquid back to a boil, then cover the pot, turn off the heat, and let sit for 25 minutes, until the water is absorbed.
3. In a large bowl, toss together the vinegar, onion, garlic and remaining salt.
4. Add the grains to the vinegar mixture while still warm; toss well. Remove the garlic and stir in the beans, tomatoes, basil and olive oil; season with black pepper to taste. Fold in the Parmesan and serve.