Separate the chicken legs from the thighs, and season with salt and pepper, to taste. Coat the pieces with the flour, shaking off any excess and reserve the leftover flour. Put the oil into a heavy bottomed skillet and heat it over medium-high heat. Add the chicken and brown it on all sides, about 5 minutes per side. Remove the chicken to a plate and add the carrots, celery, onion, mushrooms, and corn. Sprinkle with the reserved flour, and give it a good stir. Add chicken broth and stir. Nestle the chicken pieces into the vegetables and add enough water to just cover the chicken. Bring to a boil, reduce the heat, cover, and simmer for 20 minutes.
Just before the cooking time is up combine the baking mix, poultry seasoning, and milk in a small bowl. Drop the batter by tablespoons on top of the simmering chicken leaving some room between the dumplings. Cover and simmer for 20 minutes. Serve hot.