Neatly pare the base and stem of the artichokes, then cut off the tip. Rub with the lemon and place in acidulated ice water.
In a heavy Dutch oven, cook the pancetta in the olive oil over moderate heat until brown and very crisp, about 10 minutes. Remove the pancetta with a slotted spoon and reserve. Remove all but 1 teaspoon of the rendered fat from the pan and add the shallots and garlic. Cook for five minutes, stirring and picking up the brown bits from the bottom of the pan, until golden brown.
Add the carrots, bay leaves, thyme, salt, and pepper to the Dutch oven and cook until just tender, 4 to 6 minutes. Add the wine and stock and stir until it boils. Place the artichokes on the bed of vegetables, cover, and simmer over medium-low heat until tender, 20 to 30 minutes, stirring occasionally.
Add the mussels to the Dutch oven 5 to 10 minutes before the artichokes are tender and cook until they have opened. Remove any opened mussels into a large serving bowl or deep platter and cover with foil. Continue cooking the others until they open, about 5 minutes longer. Discard any unopened mussels.
If the liquid has evaporated, add 1/2 cup additional stock. Turn up the heat and reduce the liquid just to thicken slightly and adjust the seasoning with salt and pepper. Stir in the parsley and reserved pancetta. Place the artichokes and any remaining opened mussels in the serving bowl, spoon the vegetables on top, drizzle with balsamic vinegar, and serve.