Dissolve yeast in warm water in a bowl. Let stand until the yeast softens and begins to form a creamy foam, about 5 minutes.
Mix milk, mashed potatoes, shortening or butter (see Cook's Note), sugar, and salt together in a bowl; add yeast mixture and egg. Stir 2 cups flour into the milk mixture until incorporated. Stir in remaining 2 cups flour to make a soft dough.
Turn dough onto a lightly floured surface and knead until smooth and elastic, about 7 minutes. Place dough in a lightly greased bowl, turning once to coat all sides of dough with oil. Cover bowl and put in a warm place until dough has doubled in size, about 1 hour.
Punch dough down and shape into a ball. Cover the bowl and let rest for 10 minutes.
Preheat oven to 400 degrees F (200 degrees C). Grease 2 baking sheets.
Shape dough into 24 rolls and place onto prepared baking sheets; let rise until dough springs back when you touch the side of a roll with your fingertip, about 25 to 40 minutes.
Bake rolls in the preheated oven until golden brown, 10 to 12 minutes.