Season chicken generously with salt and pepper. Melt ghee in a large Dutch oven over medium heat. Add chicken and brown thoroughly on each side, 5–7 minutes per side. Transfer chicken to a plate and drain all but 1 tablespoon of fat from pot.
Add bay leaf, thyme, shallot, celery, and carrot to pot and cook for 5 minutes over medium-low heat. Add cognac and bring to simmer, scraping browned bits from bottom of pot. Add broth and water and bring to a boil. Return chicken to pot, skin side up, reduce heat, cover, and very gently simmer for 1½ hours.
Carefully remove chicken from pot and transfer to a baking sheet. Adjust oven rack to be about 7 inches from heating element and set your oven to broil. Broil chicken until skin is crispy, about 5 minutes, rotating pan as needed.
Meanwhile, bring soup to a gentle simmer and add squash. Add 1 tablespoon mounds of dumpling dough to simmering broth, you should make about 20 dumplings. Cover and gently simmer for 10 minutes.
Ladle broth with vegetables and dumplings into shallow bowls and top with pieces of chicken. Sprinkle with herbs and serve.
Combine flour, baking powder, salt, and nutmeg in a medium bowl. Add yogurt and stir until a dough forms.
NOTE: Have your butcher cut the whole chicken into 8 pieces for you, reserving the wings, neck, and spine to use in homemade chicken broth. For the recipe, visit ediblesarasota.com.