Bring a large pot of salted water to boil for your pasta. Remove broccoli florets from stems and chop into medium florets. Peel stems with a vegetable peeler and slice them into 1/2-inch segments. Use your pot of future pasta water to steam or par-boil your broccoli florets and stems until just tender, then drain if needed and set them aside. Add pasta to water and cook until al dente, or about one minute less than fully cooked. Before draining pasta, reserve a cup of pasta cooking water and set it aside. Drain pasta. Wipe out pot so that you can use it again. In the bottom of pot, melt butter and olive oil together over medium heat. Add onion and reduce to medium-low, sauteing it until tender, about 7 minutes. Add garlic and cook for another two minutes. Add steamed broccoli, salt and red or black pepper and turn the heat back up to medium-high, cooking it with the onion and garlic for a few additional minutes. Pour cream over mixture and let cook for 30 seconds. Transfer broccoli mixture and all of its creamy bits at the bottom of the pan to a blender or food processor and blend in short bursts until it’s finely chopped and a little sauce. Add the broccoli sauce back to the pot with the drained spaghetti and a splash or two of the reserved pasta water. Cook over medium-high for 1 to 2 minutes, tossing the mixture so that it evenly coats. Add more pasta water as needed to loosen the sauce. Adjust seasonings to taste, adding more salt or pepper, and scoop into a serving bowl. Shower spaghetti with grated parmesan and dig in.