Chicken and Barley Stew with Dill and Lemon

Name: Chicken and Barley Stew with Dill and Lemon
Ingredients: 1/2 cup all-purpose flour; Kosher salt; Freshly ground pepper; 2 pounds skinless, boneless chicken breasts, cut into 1 1/2-inch pieces; 1/4 cup plus 1 tablespoon vegetable oil; 2 carrots, sliced 1/4 inch thick; 2 celery ribs, sliced 1/4 inch thick; 1 large leek, white and pale green parts only, thinly sliced; 1 garlic clove, minced; 1 3/4 cups pearled barley; 6 cups chicken stock; 1 cup water; 1/4 cup chopped dill; 2 tablespoons fresh lemon juice; 1/4 cup chopped tarragon; 1/2 cup freshly grated Parmigiano-Reggiano cheese (1 1/2 ounces)