Cook pasta in a large pot of boiling water until just tender, 8 to 10 minutes or according to package directions.
Drain the pasta, reserving 1/4 cup of the cooking liquid. Place radicchio and the reserved liquid in the pot and cook over medium heat, stirring constantly, until wilted, 2 to 3 minutes. Stir in the pasta, 2 tablespoons Parmesan, oil, chicken, vinegar and pepper and continue cooking until the cheese starts to melt, 1 to 2 minutes. Serve the pasta garnished with walnuts and the remaining cheese.