Scald milk, add shortening, sugar, salt and potatoes. let cool to lukewarm. Sprinkle yeast on warm water, stir to dissolve. Combine milk mixture, yeast, 2 C flour and egg. Beat well by hand or with electric beater until batter is smooth, about 2 minutes. Gradually stir in enough extra flour a little at a time to make a soft dough that leaves the side of the bowl.
Turn onto floured board and knead until satiny and elastic - five or 10 minutes. Place in greased bowl and let rise until doubled, about an hour and a half. Punch down.
Shape in a ball and let rest for 10 minutes. Then pinch off small pieces of dough, shape into balls to half-fill greased muffin tins.
Cover and let rise until almost doubled, about an hour. Bake at 400 degrees for 10 to 12 minutes. Makes 2 dozen.