Fill a large bowl with water. Halve 1 lemon, squeeze the juice into the water and drop both halves into the bowl.
Cut off the top quarter of the artichokes with a serrated knife. Peel the tough dark green skin from the stems using a paring knife and remove the tough outer leaves until the tender inner green leaves are revealed; it should be only a couple of layers. Cut the artichokes in half lengthwise and toss into the lemon water.
Zest and juice the remaining 2 lemons and reserve. Coat a large straight-sided skillet generously with olive oil and add the garlic, anchovies and red pepper flakes. Cook over medium heat, stirring, until the garlic is very aromatic and the anchovies have dissolved, 2 to 3 minutes.
Remove the artichokes from the water and add them to the pan along with the reserved lemon zest and juice, the white wine and thyme. Season with salt and add just enough water to cover the artichokes. Bring to a boil, then reduce to a simmer. Cover the pan and cook until the artichokes are very soft, 10 to 15 minutes. Remove the lid and let the liquid reduce until it comes less than halfway up the sides of the arties, about 10 minutes.
Remove the thyme and toss in the parsley. Swirl and stir to combine. Taste for seasoning and adjust if you need to (you probably will). Put the artichokes on a rimmed serving platter or in a bowl. Use a vegetable peeler to shave the parmigiano on top.