For cheese filling, cook mushrooms in butter for about 5 minutes.
Blend in flour; stir in cream.
Add salt and cayenne.
Cook and stir until mixture becomes very thick.
Stir in cheese and cook over very low heat, stirring constantly, until cheese is melted.
Turn mixture into pie plate.
Cover, chill thoroughly, for about 1 hour.
Cut the firm cheese mixture into 6-7 equal portions.
Shape it into short sticks.
Remove skin from chicken.
Place each piece, boned side up, between 2 pieces of waxed paper.
Overlap meat where chicken breast is split.
Working out from the center, pound with mallet to form cutlets not quite 1/4 inch thick.
Peel off paper.
Sprinkle with salt.
Place a cheese stick in each chicken breast.
Tucking in the sides, roll chicken as for jelly roll.
Press to seal well.
Dust the chicken rolls with flour; dip in eggs, then roll in fine bread crumbs.
Cover and chill chicken rolls for at least 1 hour.
About 1 hour before serving time, fry rolls in deep, hot fat for 5 minutes or until crisp and golden brown; drain on paper towels.
Place in shallow baking dish and bake at 325° about 30-45 minutes.