Cook the spaghetti according to the package directions; drain and return it to the pot.
Meanwhile, heat the oil in a large skillet over high heat. Add the mushrooms, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, tossing often, until browned, 5 to 7 minutes.
Add the mushrooms, radicchio, fontina, mustard, and ¼ teaspoon each salt and pepper to the pasta and toss to combine.
Serve topped with the chives.