Rinse chicken and pat dry. Give the chicken a healthy seasoning of coarse salt and fresh black pepper before heating the butter and oil in a 4.5-qt Dutch oven (or larger) over medium-high heat. Thoroughly brown the chicken pieces in the oil and butter. Do this in batches if necessary. Remove to a plate and set aside.
Add the vegetables, bacon, bay leaf, and thyme to the pan with the chicken fat and juices. Cook for about 15 minutes over medium-low heat before returning the chicken to the pan and adding the white wine. Bring to a boil and cook for 2 minutes.
Add the stock and water, bring back up to the boil, and season with more salt and pepper. Turn off the heat under the pan, cover with a circle of parchment paper, and place the lid on top. Place the pot in the oven and cook for about 1 hour and 30 minutes.
Right before chicken is done, mix the flour, baking powder, cream, and fine salt together in a bowl until combined. Once the chicken is done cooking, remove the pan from the oven, remove and discard the parchment paper, and spoon large dollops of the flour/dumpling mixture on top of the chicken (you should end up with about 10 dumplings). Cover the pan and return to the oven for 10 minutes.
Once cooked, remove, sprinkle with fresh parsley and a bit more black pepper, and serve immediately in bowls. Be sure to spoon plenty of the juices over each portion so you end up with a soup.
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