That's my favorite combo! Pasta, cheese, and broccoli!!
It also happens to be one of my favorites which I learned way back from an Italian chef from Sicily where it originated apparently. I changed it up over the years but here's my loosely compiled recipe if your interested.
Ingredients:
9-10 oz spaghetti
1/2 head of broccoli, no stems
2 garlic cloves, sliced, or more
1/4 cup grated Parmigiana Reggiano, or more
1/2 cup of broccoli water, or more
2-3 tbsp butter or evoo
sea salt and fresh cracked pepper to taste
Method:
cook the broccoli in enough water to cover and cook until very soft, basically falling apart. Just before adding to the fry pan mush the broccoli flowerets into small pieces, a mash basically.
Cook the pasta until about 2 minutes before your preference of doneness. In a 12" pan over med-high heat add 1 tbsp of butter then add the garlic and cook for about 1 minute add the pasta and the 1/2 cup of broccoli water and continue to cook, about 1 minute later add the broccoli flowerets and combine well and continue to cook until some evaporation occurs and the pasta cooks to your liking. Turn off the heat and add the parm and butter and combine until a thick sauce is achieved, adding more water or more parm. Season with s&p and I like a lot of pepper.
Note: Originally the broccoli was bigger and added to the cooked pasta with parm and EVOO, which I did continue to do that way for many years, then I changed to this method. I like what the broccoli water does for the dish and the butter was to help emulsify the finished sauce. Also, originally I used a short pasta normally an orecchiette which worked well but I find the spaghetti and the broccoli mash clings better in the sauce from the broccoli water to the long pasta and is just better suited to my arrangement of the recipe. Of course any time we can add some chili peppers or pepperoncini is a bonus.