As taught to me by my next-door neighbor, a lovely Italian lady who never failed to let me in on her cooking secrets as a child... I like it because it doesn't have an extra step of steaming the broccoli.
Melt butter together with olive oil in large skillet with cover or dutch oven (ideally large enough to hold all of your ingredients together).
Cook garlic until no longer raw, but don't allow to brown. Add broccoli to the pan, constantly stirring, until broccoli is completely coated in the oil and garlic and pan is very hot. Add broth, and immediately cover.
Steam until broccoli is tender. Turn off heat and stir in the fresh basil.
Combine pasta and broccoli. Season with salt and pepper according to taste. Serve topped liberally with cheese.