Boil the pasta. Bring a large pot of salted water to a boil.
Cook the pasta until just al dente. Reserve 1 cup of pasta water, then drain.
Sauté the garlic. In a large skillet over medium heat, warm the olive oil and butter until melted. Add the garlic and cook for 30–60 seconds, stirring, until fragrant. Don’t let it brown.
Build the base. Pour in the chicken broth, Italian seasoning, and red pepper flakes.
Simmer for 2–3 minutes to reduce slightly. If using broccoli, add it now and simmer until crisp-tender, about 3 minutes.
Make it creamy without curdling. Lower the heat to just under a simmer. In a small bowl, whisk the Greek yogurt with 2–3 tablespoons of warm broth from the pan to temper it.
Stir the tempered yogurt back into the skillet. Keep the heat low to maintain a smooth sauce.
Add the Parmesan. Sprinkle in the grated Parmesan a little at a time, stirring until melted and smooth. If the sauce seems thick, loosen it with reserved pasta water, a few tablespoons at a time.
Add chicken and greens. Stir in the shredded rotisserie chicken and spinach (if using).
Warm through for 1–2 minutes. Season with salt and black pepper. Taste and adjust.
Toss with pasta. Add the cooked pasta to the skillet and toss until everything is coated and glossy.
A splash of pasta water helps the sauce cling. If you like, finish with lemon zest and a squeeze of lemon juice for brightness.
Serve. Top with extra Parmesan and chopped parsley. Eat hot.