Slice the radicchio into rounds, then make 2–3 lengthwise cuts to avoid long strands.
In a pan, sauté the onion with a drizzle of oil for a few minutes.
Add the radicchio, season with salt, and cook for about 5 minutes, stirring occasionally.
Bring a pot of water to a boil and cook the pasta.
Make the walnut pesto: blend walnuts, vegan cheese, garlic (optional), sage, rosemary, a pinch of salt, a drizzle of oil, and a small cup of water until smooth.
Drain the pasta al dente, reserving some cooking water.
Toss the pasta with the walnut pesto and the cooked radicchio.
Add a little pasta water and mix to make it creamy.