Cardoon is a vegetable like artichoke in that it oxidizes and discolors. Chefs that buy cardoon from us typically toss it into acidulated water (water with lemon juice) to keep it from discoloring and blanch the cardoon stalks in water that has a splash of vinegar or lemon juice until medium tender (15-20 minutes) You can peel them if you like. We don’t. Cut the cardoon stalks in 1/4 inch crescents, across the grain, like you would celery. When thinking of cardoon, keep the flavor of artichokes in mind when planning the dish.

Simplest Cardoon-Pasta preparation:
Slice and blanch cardoon. Saute onions and garlic and toss with pasta. Grate some parmesan cheese. This could benefit from some green olives as well.
Russell’s Cardoon Recipe from Russell Moore, upstairs Chef at Chez Panisse
Peel stalks of cardoons to get the bitter outer skin off. Boil in salted water until tender, that could be from 5-20 minutes. (I would cut the stalks in half so they’d fit my pans.)

Slice cooked stalks on the diagonal (like you would celery) then dress with a vinaigrette. For this dish the vinaigrette has a bit of anchovy, garlic, lemon and a small splash of red wine vinegar, and olive oil too of course. (one basic formula for vinaigrette I’ve read includes 3 parts oil to 1 part acid: lemon juice or vinegar)
Serve the cardoons this way room temperature with some hardboiled egg: either chopped or in wedges. This dish benefits from a bit of fat served with it and  the hard cooked egg are good for that contrast.