Cook the pasta according to the package directions, adding the peas during the last minute of cooking. Reserve ½ cup of the cooking water. Drain the pasta and peas and reserve the pot.
Melt the butter in the pasta pot over medium heat. Add the garlic and cook, stirring, until fragrant, about 30 seconds.
Add the cream, chicken, spinach, and ½ teaspoon each salt and pepper and cook until the chicken is warmed through and the spinach is wilted, 2 to 3 minutes.
Add the pasta and peas and toss to combine (adding a little of the reserved cooking water as needed to loosen the sauce). Serve with the Parmesan.