Stracciatella with Chicken and Barley

Name: Stracciatella with Chicken and Barley
Ingredients: 2 tablespoons extra-virgin olive oil; 2 leeks (white and light green parts), thickly sliced; 3 carrots, chopped; 8 cups low-sodium chicken broth; 1 pound skinless, boneless chicken thighs, trimmed and cut into thirds; 1/2 cup pearl barley; 1 head escarole, trimmed and roughly chopped (about 6 cups); 3 large eggs; 1/2 cup grated Parmesan cheese; Kosher salt and freshly ground pepper; Juice of 1/2 lemon; 1/4 cup chopped fresh parsley