Clean and prepare the artichokes. (To clean artichokes, fill a bowl with approximately one quart of cold water and add the juice of one lemon, plus the squeezed-out lemon halves. Peel and trim the stem of the first artichoke. Pull off any tough outer leaves and discard. Using a paring knife, trim away any tough parts around the base and stem of the artichoke. With a serrated knife, cut off the top third of the artichoke and discard.) Put cleaned artichokes in a serving bowl and toss with the lemon juice.
Heat the olive oil in a soup pot over medium-high heat. Add the potatoes, and cook until they begin to stick to the bottom of the pot, about 5 minutes. Add the leeks and shallots, and cook until softened, about 10 minutes.
Sprinkle in the garlic, thyme, bay leaf, salt, and peperoncino. Cook until the garlic is fragrant, about 1 to 2 minutes. Pour in the water and bring to a boil. Add the prepped, drained artichokes, and bring the soup to a rapid simmering. Cook, uncovered, until potatoes and artichokes are tender and the potatoes have broken down to thicken the soup, about 1 ½ hours.
Stir in the chopped parsley, and serve the soup with grated cheese