Cook the pasta according to package directions, omitting salt and fat. Drain in a colander over a bowl, reserving 3/4 cup cooking liquid.
While water for pasta comes to a boil, melt butter in a large nonstick skillet over medium-high heat. Add hazelnuts to pan; sauté for 3 minutes or until toasted and fragrant. Remove from pan with a slotted spoon. Add oil to pan, and swirl to coat. Add garlic and mushrooms to pan; sprinkle with 1/4 teaspoon salt and black pepper. Sauté mushroom mixture for 5 minutes; stir in sage. Add pasta, reserved cooking liquid, and remaining 1/4 teaspoon salt to pan; toss well to combine. Remove from heat; top with cheese, toasted hazelnuts, and chives.