Squeeze 1 lemon into a bowl with 3 cups of water. Trim the artichokes: Cut away their tough outer leaves, leaving a little of the stem intact. Trim the tough exterior with a paring knife. Using a teaspoon scrape out the purple, fuzzy choke. Rub the artichokes with the squeezed lemon half and place in the acidulated water.

Preheat the oven to 350˚F/180˚C. Peel the potatoes and cut into quarters. Toss the artichokes and potatoes with 1 cup of oil, garlic, the juice of the remaining lemon, and a little sea salt and pepper. Spread the mixture evenly in a baking pan large enough to hold the vegetables in one layer. Bake for about 1 hour, until the artichokes and potatoes are tender. About 5 minutes before removing the pan from the oven, place it under the grill, about 8 inches/20 cm from the heat source, so that the artichokes and potatoes acquire a crusty brown texture and color. Remove and serve immediately.