Bring a large pot of salted water to a boil.

Prepare the cabbage and pesto while waiting for the water to boil.

Clean the cabbage and slice into thin strands (approx 1/4" wide).

In a shallow, wide pan over high heat, add 1 tbsp extra virgin olive oil, the sliced cabbage, and 1 tsp salt.

Cook the cabbage over high heat until wilted and slightly caramelized.

Taste and adjust salt as needed.

Remove the cabbage from the heat.

Make pesto using a mortar and pestle or a food processor with the walnuts, butter, garlic cloves, 1 tsp salt, and 2 tbsp extra virgin olive oil.

Taste and adjust for salt; adjust consistency with extra virgin olive oil.

Pesto should be a thick paste.

Once water boils, add the pasta to the salted water.

Cook until al dente.

Reserve a cup of pasta water.

Drain the pasta.

In the hot pasta pot, combine pasta, cooked cabbage, and pesto.

Stir to combine.

Taste for flavor and adjust as needed.

Use the reserved pasta water to thin the sauce if necessary.

Optional: Serve with pickled onions and/or shredded parmesan cheese.