1. Bring a large pot of salted water to boil. Add broccoli and cook until soft, 7 to 9 minutes. Remove with a slotted spoon to a food processor. Add spaghetti to pot and cook until al dente, about 10 minutes or as package label directs.
2. Add olive oil and garlic to broccoli and process until smooth. Transfer to a large bowl and stir in heavy cream and Parmesan. Season with salt and pepper.
3. Drain pasta, reserving 1 cup of cooking water. Mix pasta with broccoli pesto, adding pasta water 1 Tbsp. at a time if pesto is too dry. Serve, passing additional Parmesan if desired.