Start by boiling salted water in a large pot. Cook the pasta according to package directions until al dente. Drain and set aside, reserving a cup of pasta water to adjust sauce consistency.
Heat olive oil in a skillet over medium heat. Season chicken with salt and pepper, then cook until golden and cooked through. Remove from the pan and set aside.
In the same skillet, sauté minced garlic for 1-2 minutes. Pour in the cream and chicken broth, stirring well. Let it simmer until slightly thickened.
Add Parmesan, lemon juice, and zest, stirring until smooth. Toss in the cooked pasta and spinach, stirring until the spinach wilts. Return chicken to the pan, mixing everything together.
Taste and adjust seasoning with salt, pepper, or red pepper flakes. Serve hot, garnished with extra Parmesan and lemon zest for that bright finishing touch.