Cook the pasta in salted water until al dente. Reserve a cup of pasta water, then drain.
Season the chicken with salt and pepper. Sear in a large skillet with olive oil over medium-high heat until browned and cooked through. Remove and set aside.
In the same skillet, add a touch more oil if needed. Sauté garlic and onion until translucent and fragrant, about 3–4 minutes.
Stir in red pepper flakes and cherry tomatoes. Cook until tomatoes blister and release their juices, about 2–3 minutes.
Pour in crushed tomatoes and chicken broth. Simmer for 5 minutes to meld flavors.
Lower the heat and stir in cream and Parmesan until the sauce is silky. If it’s too thick, loosen with a splash of pasta water.
Return the chicken to the pan, along with the cooked pasta and spinach. Toss until the greens wilt and everything glistens. Season to taste with salt and pepper.
Serve hot, with extra Parmesan and fresh basil if you’re feeling fancy.