Fill a large bowl with water. Cut 2 lemons in half and squeeze both lemon halves into the water, then drop in the halves.
Trim the tough dark green skin from the stem of the artichoke if there is a stem, and remove the tough outer leaves until lovely gentle spring green is revealed, it should only be a couple of layers. Cut the artichokes in half lengthwise and toss into the lemon (acidulated) water. Artichokes will turn brown before your eyes, so it is important to have the acidulated water set up before 1 leaf is removed from the artichoke.
Coat a large straight-sided saute pan over medium heat, generously with olive oil and add the smashed garlic cloves and crushed red pepper. When the garlic cloves are golden brown and aromatic, remove them. They have fulfilled their garlic destiny.Add the anchovy fillets and let them melt into a saucy consistency before adding the prepared artichokes to the pan along with the lemon halves. Pour in the white wine, add the thyme bundle and season with salt, to taste. Bring to a boil and add enough water to cover the artichokes (shouldn't be much water). Cover the pan and simmer until the artichokes are very soft, about 10 to 15 minutes. Remove the lid and let the liquid reduce until it is less than halfway up the sides of the artichokes. Remove the thyme bundle. Taste to check for seasoning and readjust, if necessary, (you probably will). Finish by tossing in the chopped parsley. Transfer the artichokes to a serving bowl and serve topped with Parmigiano-Reggiano shavings.