Mix the flour, sugar, and salt in a stand mixer. Add the eggs, yeast, potatoes, and milk. Attach the dough hook and mix on low until just combined. Increase the speed to medium and mix for 5 minutes. Place the dough in a greased bowl and cover. Allow to proof for 1 1/2 hours or doubled in size. In the meantime make the butter.
Grease a sheet pan. Turn the dough out onto a lightly oiled work surface. Divide the dough into three pieces. Working with one piece at a time, press the dough into an 8x13 rectangle about 1/4 inch thick. Cut the dough in half lengthwise. Then cut 8 strips from each half. Brush the dough with garlic butter. Roll up each strip and place them on the sheet pan. Repeat with the two remaining portions of dough. Brush the tops of the rolls with butter, cover, and proof for 30 minutes.
Preheat the oven to 350 degrees. Remove the cover and bake the rolls for 25 minutes or until browned on top. Remove from the oven and brush with the butter. Allow to cool for 10-15 minutes and serve with any remaining butter.