Fill a large pot with water. Once boiling, add a generous amount of sea salt. Add the broccoli florets and cook for 3 minutes. While florets are cooking, heat the butter and olive oil in a large sauté pan.
Add the garlic, onion, and red chili flakes and cook until translucent. Once the florets are done cooking in the water, strain and add them to the onion and garlic.
Add the milk and cook 1 minute and remove from heat. Add broccoli mixture to a food processor and blend for fifteen seconds. Scrape down the sides, add the Parmigiano Reggiano, and blend for another five seconds. Add the pasta to the water the broccoli florets were cooked in. Cook until al dente.
Once pasta is cooked, remove it directly from the pasta water pot and place it in the sauté pan with the pesto. Add a 1/4 cup of the pasta water. Slowly mix the spaghetti and the pesto, covering the strands of pasta with the sauce. Add more pasta water as necessary if you need to thin the sauce. Season with sea salt and pepper to taste. Finish pasta with grated Parmigiano Reggiano and serve immediately.