To clean the artichokes, fill a medium bowl with water and add the juice of one lemon. Remove the outer layers of the artichokes’ leaves until you reach the tender ones in the middle. Cut off the leaves of the artichoke about halfway down to the heart. Working quickly to prevent oxidation, trim off the bottom of the stem end and, with a peeler or paring knife, peel the outside of the stem until you get to the tender middle.
Cut the artichokes in half and with a small spoon scoop out the choke. Put the artichoke in the acidulated water while you complete the rest.
Meanwhile, prepare your artichoke blanching liquid. Fill a medium saucepan with water, along with a small handful of salt, a big glug of olive oil, the bay leaf, white wine, garlic, and 5 sprigs of mint. Let this liquid simmer for about 10 minutes. Taste and adust the seasoning. Add the halved, cleaned artichokes and poach for 10 to 15 minutes, until the artichokes are tender but not breaking down. Spoon the artichokes out onto a sheet tray to cool. The artichokes can be prepared a day ahead and kept cold in the refrigerator.
Once you’re ready to grill, drizzle the artichokes with olive oil, and place them cut-side down onto an oiled grill or onto a hot cast iron pan. Cook the artichokes until they’re charred and hot and transfer them to a serving platter. Top with a squeeze of fresh lemon juice, a drizzle of olive oil, a handful of the picked mint leaves, black pepper, flaky salt, and shaved pecorino.