How to Make It
Preheat the oven to 350°. Squeeze the lemon half into a bowl of cold water; add it to the water. Snap off the outer leaves of 1 artichoke. Using a sharp knife, cut off the top half and peel the base and stem. Using a melon baller or a spoon, scoop out the furry choke; halve the artichoke lengthwise and add to the lemon water. Repeat with the remaining artichokes.
Drain the artichokes; transfer to a medium saucepan. Add 1/2 cup of the oil, the garlic, mint, oregano sprigs and 1/4 cup of water; season with salt and pepper. Bring to a simmer, cover and cook over low heat until just tender, about 20 minutes. Keep warm in the saucepan.
Set a rack over a baking sheet. Heat a large cast-iron skillet. Rub the lamb all over with 2 tablespoons of the oil and season with salt and pepper. Sear 1 rack of lamb at a time over moderately high heat, turning, until golden, about 5 minutes. Transfer to the baking sheet and roast for about 30 minutes for medium-rare; an instant-read thermometer inserted in the center will register 120°. Transfer to a cutting board and let rest for 10 minutes.
Meanwhile, in a large saucepan, cover the potatoes with 2 inches of water, season with salt and bring to a boil. Cook over moderate heat until tender, about 20 minutes. Drain, reserving 1/2 cup of the cooking water. Return the potatoes to the saucepan and add the remaining 1/2 cup of oil and 1/4 cup of the reserved cooking water. Season with salt and coarsely mash with a fork; add more cooking water if necessary. Stir in the licorice powder, if using.
Transfer the artichokes to 6 plates. Carve the lamb into individual chops and serve with the artichokes and crushed purple potatoes. Garnish with oregano leaves.