Step 1Trim the ends from the radicchio. Cut the radicchio in half lengthwise, then in half crosswise. Thinly slice.
Step 2Heat the oil in a wok or large skillet over medium-high heat until hot. Add the radicchio and cook until lightly browned and tender, about 5 minutes. Lightly season with salt. Remove the radicchio from the skillet and set aside.
Step 3Bring 4 quarts of water to boil in a 5-quart saucepot. Add salt to taste. Add the pasta and cook until al dente, 12 to 14 minutes. Drain the pasta, reserving 1 cup of the cooking water.
Step 4Put the pasta in the wok or skillet in which the radicchio was cooked. Add 1/4 cup of the hot pasta water and heat, stirring, then stir in the crumbled cheese. Cook until the cheese forms a creamy sauce coating the pasta, 1 to 2 minutes. Stir in the radicchio. Season to taste with salt and pepper. Add additional pasta water if the pasta seems dry.