Bring a large pot of salted water to a boil and add the pasta, cooking it according to package instructions until it reaches al dente texture, ensuring it still has a slight bite.
During the last 4–5 minutes of cooking, add the broccoli florets into the same pot so they soften while absorbing flavor from the pasta water.
Before draining, carefully reserve about half a cup of the pasta water, which will help create a smooth and silky sauce later.
Transfer the cooked broccoli into a blender, keeping a few florets aside for texture, and add garlic, Greek yogurt, olive oil, and a small splash of reserved pasta water.
Blend the mixture until it becomes smooth, creamy, and vibrant green, adjusting the consistency by adding more pasta water if needed.
Drain the pasta and return it to the same pan over low heat, then pour the prepared broccoli sauce over it.
Gently mix everything together until the pasta is evenly coated in the creamy green sauce, making sure not to overheat to avoid curdling the yogurt.
Add parmesan cheese, salt, black pepper, and optional chili flakes, stirring until everything is well combined and flavorful.
Toss in the reserved broccoli florets for added texture and serve immediately while warm and creamy.