Cook mushrooms in butter, blend in flour, stir in sour cream (or heavy cream) Add salt & pepper Cook until mixture is very thick. Stir in cheese and cook until it melts. Turn this mixture into a pie pan or a square pan, cover and chill 1 hour. Cut cheese pieces into sticks (1 per breast)
Place chicken breasts, 1 at a time, into a Ziploc freezer bag. Pound the breast out from the center with a wooden mallet to form ¼” thick cutlets. Sprinkle meat with salt and place a cheese stick on each piece of chicken. Roll it up like a jelly roll and dust with flour, dip in egg, then crumbs. Repeat with all the rolls. Cover and chill for about an hour.
An hour before serving, fry the rolls in hot fat (375 degrees) until golden brown; drain on paper towel and place in shallow baking dish. Bake at 325 degrees for 30 – 45 minutes
I will sometimes make these ahead of time or make more than I need and freeze after the deep fry step. (Let cool first!)