1. Trim artichokes; leave stems on, and remove tough outer leaves, leaving pale-green edible ones. Neatly pare base and stem; cut off very top of artichokes. Rub with cut lemon; squeeze juice from lemons into ice water, and add lemon halves and artichokes.

2. In Dutch oven, cook pancetta in oil over medium heat until brown and crisp and fat is rendered, about 10 minutes. Remove pancetta with slotted spoon; reserve. Add shallots and garlic; cook 5 minutes, stirring and picking up brown bits from bottom, until golden.

3. Add carrots, thyme, bay leaves, salt, and pepper to Dutch oven; cook until just tender, 4 to 6 minutes. Add wine and stock, and stir until boiling. Place artichokes on bed of vegetables; cover, and simmer on medium-low heat until tender, stirring, 20 to 30 minutes. If liquid has evaporated, add 1/2 cup additional chicken stock. Raise heat, reduce liquid just to thicken slightly, and adjust seasoning with salt and pepper. You should end up with a slightly thickened, rich brown liquid. Stir in parsley and reserved pancetta.

4. Place the artichokes on a deep serving platter or bowl, and spoon the vegetables and sauce over artichokes. Drizzle with the balsamic vinegar, and serve.