Note: Prepare this at least one day before serving.
Combine the wine and 3/4 cup oil in a bowl large enough to hold the artichokes. Cut off the top half of each artichoke and discard. Peel off 3 to 4 layers of the outer leaves from each artichoke, just until the yellow leaves begin to show. Cut off the artichoke stems. At the base of each artichoke, shave off the ends of the tough outer leaves by turning the artichoke against a small, sharp knife. Trim the outside of each artichoke base until no dark green areas remain. Cut off the remaining top 1/3 of the artichoke and use a small spoon to scoop out the fibrous choke and small, purple-tipped leaves.
Make a bouquet garni by tying together the leek greens, parsley, thyme and bay leaves. Heat 1/4 cup oil in a saucepot over medium-high heat. Cook the carrots and onions until the onions are translucent, about 10 minutes, stirring frequently. Add the garlic and shallots and cook 2 to 3 minutes. Add the artichoke hearts and cook 3 more minutes. Add the bouquet garni, the stock and the anchovies. Cover the pot, reduce the heat to low, and cook until the artichokes are easily pierced with a knife, 12 to 15 minutes.