Place chicken in a Dutch oven. Add water, carrot, celery, onion, 3 sprigs of thyme, and bay leaf. Bring to a boil, cover, reduce heat to low, and simmer for 1 hour. Remove chicken from dutch oven and transfer to a bowl; set aside to cool.

Increase heat and bring the stock to a simmer. Skim off any chicken fat that appears on top of the stock and reserve in a bowl. Combine 2 to 3 tablespoons of reserved fat with flour in a small bowl; stir to make a paste, adding more flour if needed. Add the chicken fat and flour mixture to the stock. Reduce heat, simmer for 15 minutes.

Remove chicken meat from the carcass and add to the stock. Season with salt, black pepper, and cayenne pepper to taste. Continue simmering for 10 to 15 minutes.

Whisk creme fraiche, milk, 2 teaspoons of thyme leaves, and eggs together in a large bowl. Stir in self-rising flour until almost entirely incorporated; do not overmix.

Scoop large dollops of dumpling mixture on top of the chicken stock. Increase heat slightly to medium-high. Cover and simmer until dumplings appear light and fluffy, and a toothpick inserted into the center comes out clean, 10 to 15 minutes.