Heat 1 tablespoon of the oil in a large skillet over medium-high heat. Season the meat with salt and pepper and cook still pink and medium-rare, turning occasionally, 4-5 minutes.
Remove from skillet and cover to keep warm, leaving any juices in the pan. Add the radicchio and cook until wilted, 1-2 minutes. Meanwhile, cook the pasta according to package directions and drain, reserving ½ cup of the pasta cooking water.
Add the pasta to the skillet along with the beef and saffron, then toss with the remaining olive oil and cheese, adding the pasta cooking water a little at a time to loosen the sauce if desired. Season with salt and pepper and divide among bowls.