Cook pasta in well-salted boiling water until al dente. Reserve ½ cup pasta water, then drain.
Blanch or steam broccoli until bright green and tender-crisp; drain well.
In a large skillet over medium heat, melt butter. Add garlic and cook 30 seconds until fragrant.
Stir in cream and chicken broth; bring to a gentle simmer.
Add Parmesan, salt, and pepper; stir until smooth.
Add rotisserie chicken, cooked pasta, and broccoli. Toss to coat, adding reserved pasta water as needed to loosen the sauce.
Taste and adjust seasoning. Serve hot.