Bring a saucepan of water to boil. Salt generously and add pasta. Cook until al dente, tender but not soggy. Drain.
While pasta is cooking, heat a saute pan over medium heat. Add butter. When butter melts, add garlic and breadcrumbs and saute until breadcrumbs are golden and fragrant. Season with salt and pepper and add red pepper flakes (if using). Transfer breadcrumbs to a bowl.
Return pan to heat and add oil. When oil begins to shimmer, add broccoli and saute until broccoli is bright green and crisp, ~4 minutes. Add beans and pasta and stir to warm everything through, ~1 minute. Add cheese and stir until cheese is melted and pasta is evenly coated. Season with salt and pepper.
Transfer to serving plates and sprinkle breadcrumbs over the top. Enjoy!