1. Preheat oven to 350 degrees. Bring a large saucepan of water to a boil; add salt. Add pasta, and cook until al dente, 8 to 10 minutes. Drain, transfer to a medium bowl, and toss with 1 tablespoon butter. Cover, and keep warm while proceeding.

2. Spread walnuts on a baking sheet, place in oven, and toast until fragrant, about 10 minutes. When cool enough to handle, chop coarsely; set aside. Place 1/2 cup butter in a small saucepan over medium-high heat. Cook until butter begins to brown and is very fragrant, about 8 minutes. Strain, discarding solids, into a glass measuring cup or a small bowl, and set aside.

3. Heat remaining 2 tablespoons butter in a large skillet over medium heat. Add onion and garlic, and cook until they start to soften, about 2 minutes. Add sprouts leaves, sprinkle with salt, pepper, and herbs, and continue cooking until leaves are bright green and tender and onions are translucent, 3 to 5 minutes. Add pasta to skillet, and drizzle with brown butter. Toss to combine, and cook until warmed through, seasoning with salt and pepper if needed. Transfer to a serving dish, and garnish with walnuts and Parmesan cheese. Serve immediately.