In a small sauce pot, combine milk, water, and butter. Heat through on medium-low heat, until butter starts to melt (milk temperature reaches about 120° F). Take off heat.
In a bowl of your electric mixer, combine flour, sugar, salt, and yeast. Turn the mixer or low and mix to combine. Add mashed potatoes and mix until it starts to incorporate. Quickly whisk the egg into the warmed milk mixture and slowly pour the milk mixture into the flour mixture while it's mixing on "stir" setting. Add the mashed potatoes. Keep mixing on low until all combined. Raise the speed a little and beat a few seconds until smooth.
Take out the dough and place it onto a lightly floured surface. Knead the dough, adding a little bit more flour if necessary (if the dough feels very sticky), until it's elastic and smooth.
Place the dough into a lightly greased bowl and roll it so the dough gets a little greased as well.
Cover it with a towel and keep it in a warm and dry place. Let it rise for an hour to an hour and a half, until doubled in size.
Deflate the dough, take it out, and split in two even halves. Cover and let it rest for about 10 minutes.
Roll each piece into a long roll and divide each roll into 12 pieces. Roll them into smooth balls and place them onto a lightly greased baking pan. (I used 2 13x9 baking pans with 12 rolls in each pan.)Notes:- you can also divide them into 16-18 pieces if you want larger rolls. - if you want the rolls that sit tight together that you have to pull apart, place more rolls into the baking dish, closer together. If you want rolls that are more individual and bakes on all sides, place them into a larger baking dish and spaced apart about 2 inches.
Cover and let them raise for about 30-45 minutes, until doubled in size.
Preheat the oven to 375° and bake the rolls for about 18-20 minutes, until golden brown and feel set. Note: you can also take internal temperature of the center roll, in the middle. Dinner rolls are done when they reach 190° internal temperature.