Pasta: Bring heavily salted water to a boil and cook according to package instructions until pasta is al dente. Reserve ½ cup of water that the pasta cooked in.
Cut Radicchio: Meanwhile, trim root from each radicchio, then cut in half lengthwise. Lay flatside down, then cut into strips. To help remove bitterness at this point, you can soak the leaves in water for 15 minutes.
Saute: Heat oil in a large saute pan over medium heat. Add the garlic and sliced radicchio. Cook until radicchio wilts down, about 5 minutes.
Flavor: Add spices, goat cheese, and reserved ½ cup of pasta water. Stir to evenly combine.
Serve: Stir radicchio mixture into drained pasta and serve warm. Top generously with fresh grated parmesan and balsamic oil (or reduction).