How to Make It
Fill a bowl with water and squeeze the lemon halves into it. Working with 1 artichoke at a time, snap off the outer leaves. Trim the stem and cut off the top two-thirds of the leaves. Thinly slice the artichoke lengthwise and add to the water. Repeat with the remaining baby artichokes.
In a large skillet, heat 2 tablespoons of the oil until shimmering. Add the onion and cook over moderate heat, stirring occasionally, until lightly browned, 15 minutes. Transfer to a plate and wipe out the skillet.
Heat 2 tablespoons of the oil in the skillet until shimmering. Season the livers with salt and pepper; add them to the pan in a single layer along with the rosemary. Cook over moderate heat until lightly browned on the bottom, about 1 minute. Return the onion to the pan and cook, turning the livers until they're just pink in the center, about 1 minute. Add the vinegar, lemon juice and the remaining 2 tablespoons of oil to the skillet and cook over high heat until reduced by half, about 1 minute. Discard the rosemary sprigs.
Drain the artichokes and pat thoroughly dry. Arrange the lettuce and artichokes on a platter. Spoon the chicken livers and pan sauce on top and serve at once.