Bring a large pot of salted water to a boil and cook the pasta according to package instructions. During the last two minutes, add broccoli florets to the same pot. Drain everything together and set aside.
Heat olive oil in a large skillet over medium heat. Add chicken pieces, season with salt and pepper, and cook until golden and fully cooked through. Remove chicken from the skillet and keep it warm.
In the same skillet, add garlic and cook briefly until fragrant. Pour in chicken broth, lemon juice, and lemon zest, then let it simmer gently to reduce slightly and concentrate flavor.
Lower the heat and stir in Greek yogurt until smooth. This creates a creamy texture without heaviness. Add Parmesan cheese slowly, stirring until fully melted and combined.
Return cooked chicken to the skillet, followed by pasta and broccoli. Toss gently until everything is coated evenly in the lemony sauce. Adjust seasoning if needed.
Remove from heat and sprinkle with fresh parsley and red pepper flakes if using. Let it rest for a minute so flavors can settle before serving.