Pat the chicken breasts dry with kitchen paper then dip in the flour, shaking off any excess.
Heat a large, deep, lidded frying pan or casserole for a couple of minutes over a moderately high heat. Add one tablespoon of oil then, when that is hot, add half the butter.
Place the chicken breasts in the pan, skin-side down and fry for 2-3 minutes, or until the skin is nicely browned. Turn the breasts over, turn the heat down and fry the other side for 1-2 minutes. Remove the chicken breasts to a plate, discard the fat in the pan and wipe it clean.
Return the pan to a moderate heat and add the remaining oil and butter. Add the onion, carrot and celery, stir well, cover the pan and cook for five minutes, or until the vegetables are beginning to soften.
Stir in the thyme and any leftover flour. Add the chicken stock and cider, bring to the boil and add the sliced apples. Turn the heat down and return the chicken pieces to the pan, spooning the vegetables and apple over them. Replace the lid and simmer for about 35-40 minutes, turning the chicken breasts over half way through. Add a little extra cider or chicken stock if needed. Check the seasoning, adding extra salt or pepper to taste.