How to Make It
In a bowl, combine the anchovies, garlic, parsley, mint and 5 tablespoons of the oil. Season the stuffing with salt and pepper.
Dip the cut side of one of the lemon halves in salt. Working with one artichoke at a time, snap off the tough outer leaves. Using a small, sharp knife, cut off the top third of the artichoke and trim and peel the base. Using a 1/4-inch melon baller, scoop out the hairy choke. Squeeze a little of the juice from the unsalted lemon half into the cavity. Rub the exterior of the artichoke all over with the salted lemon half. Repeat with the remaining artichokes. Using a small spoon, fill the artichokes with the anchovy stuffing.
In a large, deep skillet, heat the remaining 5 tablespoons of olive oil. Set the stuffed artichokes in the skillet, stem side up, and cook over moderately high heat until nicely browned, about 5 minutes.
Add the water to the skillet and bring to a boil. Reduce the heat to moderate, cover and cook for about 15 minutes, until the artichokes are tender when pierced with a knife. Transfer the artichokes to a platter.
Raise the heat to high and boil the cooking liquid until reduced to a syrupy glaze. Season with salt and pepper and pour the glaze over the artichokes. Serve at room temperature.