Bring a large pot of generously salted water to a boil (salty like the sea). Add ziti and cook al dente. In the last 3 minutes, add broccoli and boil for 2-3 minutes. Strain together and set aside.
Meanwhile, heat 1 tablespoon of butter over medium heat in a large pan or cast-iron skillet. Add chicken, season with salt and pepper, and fully cook. About 5-7 minutes. Remove chicken and transfer to a bowl, leaving juices behind in the pan. Cover chicken with foil to keep warm.
Melt the remaining butter and add onion to the skillet. Simmer until onions are translucent, about 3-4 minutes (keep them moving to prevent burning). Add garlic and cook until fragrant (1 minute). Add chicken broth, flour, cream, and red pepper flakes. Whisk and bring to a boil and simmer for about 7 minutes until slightly reduced and thickened (whisk occasionally).
Add parmesan cheese and stir/whisk until incorporated. If the sauce is still too thin, whisk in an additional tablespoon of flour, it should thicken in less than one minute.
Return pasta and broccoli to the skillet and toss in the creamy sauce. Add chicken and continue to toss until everything is well-coated.
Serve with crushed red pepper flakes and additional parmesan cheese!