Make mashed potatoes. Cut potatoes into 4-5 pieces and put in a pan with water. Boil for 20-25 minutes, add salt after boiling. Drain the water. Mash the potatoes into a puree without lumps, add a piece of butter. Let the puree cool completely.
Form the patties. Add the egg, flour, salt and pepper to taste to the mashed potatoes. Mix thoroughly. Form the patties. Coat the patties in breadcrumbs or flour on all sides.
Prepare the gravy. Put the butter in a saucepan and melt it. Add the flour, fry over medium heat for 3-4 minutes, stirring constantly. Pour in the hot broth or water, stir with a whisk. The main thing is that the mass is without lumps. Add the sour cream, stir thoroughly. Salt and pepper. Bring the sauce to a boil, reduce the heat to low and cook for another 2-3 minutes.
In another frying pan, fry the potato cutlets. Heat the frying pan with vegetable oil, fry the cutlets for 3-4 minutes on each side until golden brown. Place the finished cutlets on a plate, pour sour cream sauce over them.