Chicken and Dumpling Soup

Name: Chicken and Dumpling Soup
Ingredients: One 3 1/2-pound whole chicken; 1 teaspoon whole black peppercorns plus freshly ground black pepper; 2 bay leaves; 2 cloves garlic, crushed and peeled; 2 sprigs fresh thyme; 3 large carrots, 1 cut into large chunks and 2 thinly sliced; 3 large celery stalks, 1 cut into large chunks and 2 thinly sliced; 2 medium yellow onions, 1 quartered and 1 chopped; 2 medium parsnips, 1 cut into large chunks and 1 thinly sliced; 6 sprigs fresh dill, 3 left whole and 3 chopped; 6 sprigs fresh flat-leaf parsley, 3 left whole and 3 chopped, plus more for serving; 1 tablespoon kosher salt, plus more as needed; 6 tablespoons unsalted butter; 1 1/2 cups all-purpose flour; 1 1/2 teaspoons kosher salt; 1 1/2 teaspoons sugar; A few fresh grinds whole nutmeg; 3 large eggs