For the broth:
2 chickens (4 pounds each), backs removed and cut into 8 pieces each
1 large onion, coarsely chopped
6 cups water
2 bay leaves
1 1/2 teaspoons salt
For the dumplings:
2 cups all-purpose flour
4 teaspoons baking powder
1 teaspoon salt
3 tablespoons butter
1 cup milk
7 tablespoons butter
10 tablespoons all-purpose flour
1/4 cup dry white wine
2 teaspoons cajun seasoning
1 teaspoon fresh thyme leaves
3 ribs celery, cut into 1/2-inch pieces on the diagonal
4 carrots, peeled and cut into 1/2-inch pieces on the diagonal (optional)
1 large onion, cut into 1-inch pieces
1/4 cup heavy cream (optional)
1/2 cup frozen green peas, thawed (optional)
Salt and freshly ground black pepper
3 tablespoons chopped fresh parsley leaves
Cut chicken backs, necks and wings into 1-inch pieces.
For the broth: in a large pot, combine the chicken, onions, water, bay leaves and salt. Bring to a boil. Reduce heat to low and simmer, partially covered, until the chicken is tender, about 45 minutes. Remove the chicken pieces and set aside. Discard the chicken backs, necks, and wings. When the chicken pieces have cooled, remove the meat from the bones in chunks and set aside. Discard the skin and bones. Strain the broth and discard the vegetables.
Prepare the dumplings: In a medium bowl, mix the flour, baking powder, and salt together. In a small saucepan over low heat, bring the butter and milk to a simmer. Stir the butter and milk mixture into the dry ingredients with a fork and stir until the mixture just comes together. On a lightly floured surface, roll the dough out to a 1/8-inch thickness. Cut into long strips 1-inch wide. Transfer to a large plate or baking sheet and cover with plastic wrap. Refrigerate until ready to use.
In the cleaned Dutch oven, melt the butter over medium heat. Whisk in the flour and cook until golden-brown, 3 to 4 minutes. While whisking, add the wine, the reserved chicken broth, cajun seasoning, and thyme. Cook until thickened, about 5 minutes, then add the celery, carrots, and onion. Cook until the vegetables are tender-crisp, about 15 minutes. Add the reserved chicken meat, heavy cream, and peas. Season with salt (if necessary), and add black pepper, to taste.
Place the dumplings on top of the chicken mixture and gently stir into the hot liquid. Cover and simmer until the dumplings are cooked through, about 10 minutes. Gently stir in the parsley. Serve in large soup bowls.