Bring a medium pot of salted water to a boil and add the Brussels sprouts. Cook until crisp-tender, about 5 minutes. Drain.

Working in two batches melt 3 tablespoons of the butter over medium-high heat in a large saucepan. Add half the shallots and half the garlic and cook for 1 minute. Add half the sprouts in a single layer and cook until golden brown, about 5 minutes, stirring as needed.  Season with 1/2 teaspoon salt and 1/4 teaspoon of the black pepper. Remove from the pan and set aside. Repeat with 3 more tablespoons of butter and the remaining shallots, garlic, Brussels sprouts, salt and pepper.

Melt the remaining 6 tablespoons butter in the pan and continue to cook until  golden brown.  Add the walnuts, cook for 30 seconds, then add the lemon juice.  Swirl the pan to combine the lemon juice with the browned butter.  Add all the Brussels sprouts to the pan and continue, stirring as needed to cook until warmed through.  Season with more salt and pepper as needed.

Transfer to a serving bowl and sprinkle with the Parmesan cheese. Serve immediately.