Heat 2 tablespoons olive oil in a large skillet over medium-high heat until shimmering. Add prosciutto and cook until crisp, about 3 to 4 minutes, stirring occasionally. Transfer to a small bowl.
Add remaining 2 tablespoons oil to the same skillet and heat until shimmering. Add mushrooms and a pinch of salt and cook until golden brown, about 5 minutes. Transfer to the bowl with the prosciutto.
Meanwhile, bring 4 quarts water and 1 tablespoon salt to boil in a large pot. Add pasta and cook 9 minutes. Add asparagus and cook 3 minutes longer, 12 minutes total.
Drain pasta and asparagus (do not rinse). Return to pot. Stir in butter, 8 ounces Roth's Private Reserve cheese, prosciutto, and mushrooms and toss until uniformly combined. Season to taste with salt and pepper.
Transfer to a large serving bowl or individual plates. Garnish with more Roth's cheese and fresh chives.