Fill an All-Clad 3 Qt. Mixing Bowl half full with water. Cut the lemons in half and squeeze the juice of three of the lemons into the water.
Proceed with trimming the artichokes by grasping the artichoke by the stem with one hand and using your other hand to carefully pull the leaves back until they snap off. Begin snapping the small leaves at the stem and work your way around until the tender pale yellow leaves are exposed. Repeat with the remaining artichokes. Next, cut the stems off where they meet the base and rub the exposed cut with the cut lemon to prevent it from discoloring. Turn the artichoke on its side and cut the yellow leaves from the base where they meet the heart. Discard the leaves. Hold the artichoke heart in your hand with the stem side down and use a sharp paring knife to trim away the tough outer green layer from the sides of the heart. It helps to turn the artichoke with your hand while keeping the knife in place. Turn the artichoke over so that the stem faces up and trim away the dark green from the base at roughly, a 45° angle until only light colored flesh remains. Rub the artichoke heart thoroughly with the lemon. Use a sharp spoon or melon baller to scrape out the fuzzy choke from the center of the heart.
Immediately immerse the artichoke into the lemon water.
Repeat the trimming process with the remaining artichokes.
Heat the All-Clad TK 8 Qt. Rondeau over medium heat and add the olive oil. Add the onions and a teaspoon of kosher salt.
Sweat the onions, stirring frequently until they begin to wilt, about 3 minutes. Stir in the garlic, carrots, and another teaspoon of salt, then cover the rondeau and cook over low heat for 2 minutes.
Drain the artichokes, then arrange them stem side up over the vegetables and cook for about 30 seconds. Turn the artichokes over and cook for another 30 seconds. Add the white wine and quickly bring to a simmer and cook for 1 minute. Add the chicken stock to cover the artichokes by about ½ inch. Season the liquid well with salt and add the bouquet garni. Bring the liquid to a simmer, then lay a clean, lint-free dish towel or a couple of layers of cheese cloth over the surface to keep the 3 artichokes submerged. Cook the artichokes for about 35 minutes or until they are tender when pierced with a paring knife.
Remove the rondeau from the heat. Stir in the minced anchovies and allow the artichokes to cool in the liquid. At this point the artichokes can be stored in the liquid, refrigerated, for a couple of days.
When you are ready to serve, discard the bouquet garni and remove the artichokes from the liquid. Cut each base into six wedges. Drain most of the liquid from the vegetables and reserve for another use such as a soup or a pasta sauce. Combine the artichokes with the vegetables and minced parsley and reheat or serve at room temperature.