Prepare the chicken: In a shallow dish, combine the flour with 1 teaspoon salt and ½ teaspoon pepper. Dredge the chicken pieces in the seasoned flour. In a large skillet or sauté pan, heat 1 tablespoon of the oil over medium-high heat, then add the chicken pieces. Cook until thoroughly browned, about 5 minutes on each side; transfer to a 4-quart slow cooker.

Meanwhile, slice the leeks into ½-inch rounds and wash and drain them well. Cut the fennel in half vertically, cut out the tough core, then cut the fennel crosswise into ½-inch slices. Return the skillet to medium-high heat and add the remaining 1 tablespoon oil. Add the leeks, fennel, and 2 tablespoons of the dredging flour. Cook, stirring frequently and scraping up the browned bits from cooking the chicken, for about 5 minutes, until the vegetables are slightly softened and the flour is cooked and golden brown. Add 1½ cups of water and bring to a boil, stirring.

Pour the vegetables and liquid into the cooker over the chicken. Cook, covered, on the low setting for 3½ to 4 hours, until the vegetables are soft and the chicken is falling off the bone and tender.

If the liquid is too runny, spoon a ¼ cup of it into a measuring cup and whisk in 1 tablespoon flour, then pour it back into the cooker. Stir in the thyme and minced parsley and add more salt and pepper to taste, as necessary.

Prepare the dumplings: In a medium bowl, combine the flour, baking powder, salt, and parsley with a fork, then add the milk. Stir with the fork until the dry ingredients are just combined and moistened—do not over mix the dough. Use a large spoon to gently scoop ½-cup dollops of dumpling dough onto the surface of the gravy in the cooker. Sprinkle the tops of the dumplings with the pepper and salt. Cook, uncovered, on the high setting, for 5 minutes, then replace the lid and cook for 25 to 30 minutes longer, until the dumplings are cooked through and fluffy; a toothpick inserted in the center of a dumpling will come out clean. Serve in large bowls, garnished with the parsley sprigs and reserved fennel fronds.