In a large bowl or the bowl of a stand mixer, dissolve the yeast in the warm water. In a small microwave safe bowl or measuring cup, add the milk, mashed potatoes, and 4 tablespoons butter. Heat on high for one minute, stir, then heat for an additional 30-60 seconds, until the butter is melted and the milk is warm but not hot.
Add the milk mixture, sugar, salt, and egg to the yeast mixture. Add two cups of flour, and mix to combine. Mix in additional flour, 1/4 cup at a time, to form a soft dough that pulls away from the sides of the bowl. Knead until smooth and elastic, about 6-8 minutes. Shape the dough into a ball, place in a greased bowl, cover, and let rise for about one hour, until doubled.
After one hour, gently punch the dough down. Preheat the oven to 400. Divide the dough into 24 pieces and shape each piece into a ball. Place the balls of dough on a greased or parchment lined baking sheet. Cover and let rise for 30-60 minutes, until the dough springs back when touched lightly.
Bake 10-12 minutes, until golden brown. Brush with the remaining 2 tablespoons of butter. Serve warm.