Fill a medium bowl with water and squeeze 1 lemon into it.

Working 1 at a time, trim the artichoke stems to 1-inch in length. With a paring knife, trim away the tough stem exterior. (As you work, rub the cut artichoke surfaces with a lemon quarter to prevent discoloration.) Break off the leaves near the base and with the artichoke held at an angle on the cutting board, cut away the leaves to leave a cone shape. Cut the leaves to within 1-inch of the heart, and with a spoon, scrape out the tough hairy choke, leaving the tender heart and stem. Cut in 1/2 lengthwise. Place in the lemon water to prevent discoloration while preparing the remaining artichokes.

In a large saucepan, heat 1/4 cup of the oil over medium-high heat. Add the onions and cook, stirring, until soft, about 4 minutes. Add the garlic and cook, stirring, for 30 seconds. Add the artichokes and brown on both sides. Add the greens, water, dill, lemon juice, salt, and pepper, and stir to combine. Reduce the heat to low, cover, and cook, stirring, occasionally, until the artichokes are tender, 20 to 25 minutes.

Remove from the heat and adjust the seasoning, to taste. Place in a serving dish and drizzle with the remaining 1/4 cup olive oil. Serve.