Pat the chicken dry with paper towels. Place 1/4 cup of the flour in a large bowl or sealable plastic bag and season with salt and pepper. Add the chicken in batches and stir or shake to coat thoroughly with the seasoned flour.

In a large Dutch oven over medium-high heat, warm 3 Tbs. of the olive oil. Working in batches to avoid crowding, add the chicken and brown on all sides, 4 to 5 minutes per batch. Transfer to a bowl and set aside.

Reduce the heat to medium and warm the remaining 1 Tbs. oil in the pot. Add the leeks and onions and saute until lightly browned, 5 to 7 minutes. Add the carrots, celery and apples and saute until slightly softened, 3 to 5 minutes more. Add 2 Tbs. of the dill, the stock and apple juice and bring to a simmer. Return the chicken and any accumulated juices to the pot, cover and cook until the chicken is cooked through and the juices run clear when a thigh is pierced with a knife, about 15 minutes. Season with salt and pepper.

Meanwhile, make the dumplings: In a bowl, stir together the remaining 1 1/4 cups flour, the cornmeal, baking powder, the 1/2 tsp. salt and the remaining 1 Tbs. dill. Using a pastry blender or 2 knives, cut in the shortening. Add the milk, stirring with a fork until the dough holds together in a rough mass.

Using an ice cream scoop, scoop up spoonfuls of the dumpling batter and arrange over the chicken in the simmering broth (you should have 6 to 8 dumplings). Cover and cook, spooning some of the broth over the dumplings once or twice, until a skewer inserted into a dumpling comes out clean, 15 to 18 minutes.

Ladle the stew into warmed bowls, dividing the chicken and dumplings among them. Serve immediately