Heat 2 tbsp of the oil in a large, heavy-based pan over a medium-high heat. Season the flour generously with salt and pepper, then evenly dust the chicken pieces. Fry them in batches until golden brown on all sides. Remove the chicken from the pan and put to one side.
Turn the heat down a little, and add the remaining oil, garlic, herbs, carrot and celery, with a pinch of salt. Fry for at least 5 minutes, adding the anchovies after 3 minutes, until the veg is softened and slightly golden and the anchovies have melted in.
Pour in the tomatoes and vinegar and scrape the bottom of the pan to remove any stuck bits. Return the meat to the pan and pour over the stock. Bring to a simmer, cover, and cook gently for 45 minutes, until the meat is falling off the bone.
Season to taste and serve with a green salad, spinach, or green beans.