1. Rinse the chicken under cold running water, removing any excess fat. Pat dry with paper towels.
2. Heat the oil in a Dutch oven or large, heavy pot over medium-high heat. When it just begins to smoke, add only enough chicken drumsticks and thighs to fit into the skillet without touching and sauté until well browned on all sides, 8 to 10 minutes total. Using tongs, transfer the pieces to a platter and set aside while you sauté the remaining pieces in batches, as necessary.
3. When all the chicken has been removed from the pot, reduce the heat a bit and sprinkle the flour over the drippings, whisking rapidly to blend. Cook the roux, whisking constantly, until it achieves a peanut-buttery color, 5 to 6 minutes. Watch carefully; you don’t want to burn the roux. As soon as the roux is the right color, add the chopped vegetables along with the ham, bacon, and garlic; be careful, because steam will be created on contact with the very hot roux. Reduce the heat to very low and sauté the mixture gently, stirring frequently, until the vegetables start to soften and give off a little liquid, about 5 minutes.
4. Add the bay leaves, salt, cayenne, black and white peppers, and thyme and stir well to mix. Add 4 cups chicken stock, stirring well to mix, then return the chicken legs to the casserole. Add more stock, if necessary, to cover the chicken and bring to a boil over high heat, stirring occasionally. Reduce the heat to low and simmer, uncovered, until the chicken is very tender and almost but not quite falling off the bone, 25 minutes.
5. Using a slotted spoon, transfer the chicken legs to a platter and set them aside. Increase the heat under the pot to high and cook the sauce, uncovered, until it’s reduced to a light sauce consistency, 10 to 15 minutes. Return the chicken to the pot and cook until heated through, 3 to 5 minutes. Remove the pot from the heat, stir in the filé powder, and let stand for 5 minutes before transferring to a serving dish.