Peel and chop the onion. Wash and slice the mushrooms. Cut the kidneys in half lengthways. Remove the white fatty core and cut the kidneys into quarters.
Melt the butter and oil in a saucepan over a moderate heat, and fry the onion gently for 2-3 minutes to soften. Add the mushrooms and stir. Add the kidneys, flour, salt and pepper, and continue to cook for another 3-5 minutes until the kidneys are browned on all sides.
Remove the pan from the heat and gradually stir in the milk. Return to the heat and slowly bring the sauce to the boil, stirring gently all the time as the sauce thickens. Lower the heat, cover the pan and simmer very gently for 10-15 minutes, until the kidneys are cooked. Cook the rice (while the kidneys are simmering) in boiling, salted water. Stir the yoghurt into the kidney sauce. Drain the rice and serve hot with the kidneys.