Cook pasta in 6-8 quarts of rapidly boiling, salted water until al dente (about 8-10 minutes). Drain and transfer to large warmed bowl, toss with 2 tablespoons of balsamic and 2 tablespoons of Basil olive oil.
Cook broccoli florets for 2 minutes in just enough boiling water to cover, drain and set aside.
Heat remaining olive oil in large heavy skillet, add chicken and cook until no longer pink. Add broccoli and sauté, for 2 minutes, or until heated through.
Toss chicken and broccoli mixture with pasta. Mix in tomatoes and basil, reserving some basil for garnish. Top with cheese. Drizzle on more olive oil and balsamic to taste. Garnish with julienned basil. Serve hot or at room temperature.
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