1/2 C freshly grated Parmigiano Reggiano, plus more for finishing
Freshly ground black pepper
Extra virgin olive oil for finishing
Fill a large pot with water. Once boiling, season the water with a generous amount of sea salt. Add the broccoli florets and cook for 3-5 minutes, depending on how large your florets are. While the florets are cooking, heat the butter and olive oil in a sauté pan over medium heat.
Add the onion, garlic, and red chili flakes (if using) and cook until translucent. Once the florets are done cooking in the water, strain and add them to the onion and garlic mixture.
Add the milk, cook for a minute, and then remove from the heat. Pour the broccoli mixture into a food processor and blend for fifteen seconds. Scrape down the sides, add the freshly grated Parmigiano Reggiano, and blend for another five seconds. Pour the broccoli pesto into a large mixing bowl and set aside.
Add the pasta to the water the broccoli florets were cooked in. Cook until al dente. Once the pasta is cooked, remove it directly from the pasta water pot and place it in the bowl with the pesto. Add an 1/8 cup of the pasta water. Slowly mix the spaghetti and the pesto, covering the strands of pasta with the sauce. Keep adding more pasta water, a little at a time, if you need to thin the sauce. Taste and season with sea salt if necessary. Divide the pasta into individual bowls. Finish the dish with some freshly ground black pepper, grated Parmigiano Reggiano, and a drizzle of extra virgin olive oil.