In a large frying pan, in 2 tablespoons vegetable oil (or ghee) fry potatoes until brown. Drain and reserve the potatoes.
Add remaining oil to the pan and fry onion, garlic and ginger until onion is soft and golden. Add chili, pepper, turmeric, cumin, salt and the tomatoes. Fry, stirring constantly for 5 minutes.
Add yogurt, mint, cardamom and cinnamon stick. Cover and cook over low heat, stirring occasionally until the tomatoes are cooked to a pulp. It may be necessary to add a little hot water if the mixture becomes too dry and starts to stick to the pan.
When the mixture is thick and smooth, add the chicken pieces and stir well to coat them with the spice mixture. Cover and cook over very low heat until the chicken is tender, approximately 35 to 45 minutes. There should only be a little very thick gravy left when chicken is finished cooking. If necessary cook uncovered for a few minutes to reduce the gravy.
Wash rice well and drain in colander for at least 30 minutes.
In a large deep frying pan or saucepan, heat vegetable oil (or ghee) and fry the onions until they are golden. Add saffron, cardamom, cloves, cinnamon stick, ginger and rice. Stir continuously until the rice is coated with the spices.
In a medium-size pot, heat the chicken stock and salt. When the mixture is hot pour it over the rice and stir well. Add the chicken mixture and the potatoes; gently mix them into the rice. Bring to boil.
Cover the saucepan tightly, turn heat to very low and steam for 20 minutes. Do not lift lid or stir while cooking. Spoon biryani onto a warm serving dish.