Here is a link for how to clean the baby artichokes. They either don’t yet have a hairy choke, or it’s young and edible. You just need to remove the tougher outer leaves until you reach the paler ones, cut off the pointy tip and the cut stem, peel the outside of the stem, and then you can eat the rest. You will be shocked by the amount of waste, but you will forget about it when enjoying your trimmed and cooked baby artichokes. 

https://www.saveur.com/article/Techniques/Trimming-Baby-Artichokes/

Once trimmed, you can put them in acidulated water or rub them with a cut lemon. If you are frying them (cooking in oil), you should rub them with lemon; otherwise, the water will pop and splatter in the oil. You don’t need to hold them in acidulated water, but it will inhibit them from discoloring.

TO GRILL:

I got some of these artichokes at the market last week and cleaned them, halved them, held them in acidulated water, then blanched them briefly in heavily salted water, drained them well, tossed them with olive oil, pounded garlic and lemon zest, and then grilled them. When they were done grilling, I threw them back in the bowl they had marinated in with the seasoned olive oil and added a squeeze of lemon. You can keep it this simple or add more flavor to your blanching water (peppercorns, chili, onion, lemon zest, herbs, etc.…)

PAN BRAISE:

Once you’ve cleaned them (and held them in acidulated water, then drained and dried them), you can also sear them in olive oil, then once they are golden add sliced shallots, garlic or green garlic, maybe some thyme and chili flakes, and let those soften. If you cut the artichokes in to quarters they could be done at this point. Or if they need more time and you want them a little juicier, deglaze the pan with a little white wine and water/broth. If you add a very small amount of wine and water/broth, the liquid will reduce and concentrate quickly and steam the artichokes to finish them. Or add even a little more water/broth and braise them until they are tender.

SHAVE RAW:

Trim the artichokes, shave thin on a mandolin and toss with olive oil, lemon juice, and shaved parmesan.

These are just some basic methods to get you started. More to come as the season continues...