Combine onion slices and 4 cups ice water in a large bowl; let stand 30 minutes. Drain and pat dry with paper towels.
Whisk together vinegar, olive oil, 2 tablespoons water, and next 4 ingredients. Stir in basil.
Layer half of tomato slices in a shallow dish. Top with half of onion slices. Drizzle with half of dressing. Repeat with remaining tomato slices, onion slices, and dressing. Cover and chill 30 minutes.
Sprinkle with Parmesan cheese curls, if desired, and serve with bread, if desired. Serve at room temperature.
Note: Recipe may be prepared up to 3 hours ahead.
Per serving (with bread and Parmesan): Calories 203 (36% from fat); Carb 28g; Fiber 2.2; Fat 7.8g; Sodium 414mg.