Put broth, 2 cups water, 2 tsp. salt, bay leaves, half of the thyme sprigs, the garlic, and peppercorns in a large pot. Halve 1 of the lemons, squeeze juice from both halves into pot, and drop in juiced rinds.
Working with 1 artichoke at a time, snap off outer leaves (about half of it; be ruthless) down to the yellowish core. Trim 1/2 in. from leaf points and trim stem to about 1/2 in. Drop into broth mixture.
Bring artichokes in broth mixture almost to a boil. Reduce heat and simmer, covered, until leaves are tender when pierced with a sharp knife, 20 to 30 minutes. Let cool to room temperature, covered; then chill overnight and up to 3 days.
Let artichokes come to room temperature (about 1 hour). Drain artichokes, cut them in half lengthwise, and remove cooked thyme sprigs. Divide artichokes and mozzarella balls among serving plates, drizzle with olive oil, and sprinkle with salt and pepper. With a zester or a Microplane, shred about 1 tsp. zest from remaining lemon and scatter over plates along with remaining thyme sprigs. Serve with red-pepper pine nut aioli and crusty bread.
Note: Nutritional analysis is per 3-artichoke serving.