Combine 1 cup flour, yeast, sugar, and salt in the bowl of a stand mixer fitted with the paddle attachment.
Combine water, milk and butter in a microwave safe measuring cup. Heat to between 120°-130°F (about 1 minute in my microwave). It’s best if you can use an instant read thermometer, but if you don't have one, heat your water to very hot bathwater temperature. Stir in potato flakes.
Pour the liquid ingredients into the dry ingredients and mix with the dough hook for 1 minute on medium speed. Add egg and 1 cup flour and mix another 1 minute. Add a third cup of flour and mix until your dough comes together to a soft dough. It will be slightly sticky to the touch. If needed, add up to 1/4 cup more flour.
Turn out the dough onto a floured surface and dust it with flour. Knead until the dough comes together in a ball and the dough bounces back when pressed with two fingers. This will take about 4-6 minutes (longer if you've done this process by hand). Cover and let the dough rest for 10 minutes.
Divide dough into 12 equal pieces. Roll each into a ball and place into a 9x13-inch pan that's been sprayed with nonstick cooking spray. Cover and let rise until doubled, about 30 minutes to 1 hour.
Preheat the oven to 375°F. Bake for 20-28 minutes, or until rolls are golden. Be sure to rotate the pan if needed partway through baking to ensure the rolls get evenly golden.
Brush hot rolls with melted butter, if desired. Enjoy within 2 days.