Put the breadcrumbs in a medium bowl and moisten with 1/2 Tbs. water. Mix in the Parmigiano, parsley, 3/4 tsp. salt, and 1/2 tsp. pepper. Add the ground chicken and egg and mix until just combined. Scoop out 1 Tbs. of the chicken mixture and, with damp hands, roll it into a 3/4-inch meatball. Transfer to a plate and shape the remaining meatballs.
Heat the oil in a 5-quart pot over medium heat. Add the onion, carrot, celery, and 1 tsp. salt; cook until tender, 3 to 4 minutes. Add the chicken broth, rosemary, and 2 cups of water; bring to a boil over medium-high heat. Gently add the meatballs, reduce the heat to medium low to maintain a gentle simmer, and cook for 10 minutes.
Add the escarole and continue to simmer until the meatballs are cooked through and the escarole is wilted, about 5 minutes. Discard the rosemary and season to taste with salt and pepper. Serve sprinkled with grated Parmigiano.