WARM OVEN Adjust oven rack to middle position and heat oven to 200 degrees. When oven reaches 200 degrees, turn it off. Line baking sheet with parchment paper. Grease large bowl.
MAKE DOUGH Whisk water, oil, sugar, and yeast in liquid measuring cup until yeast dissolves. In bowl of stand mixer fitted with dough hook, mix flour, potato flakes, and 1½ teaspoons salt until combined. With mixer on low, add water mixture in steady stream and mix until dough comes together, about 1 minute. Increase speed to medium and knead until dough is smooth and comes away from sides of bowl, about 6 minutes.
LET RISE Turn dough onto lightly floured counter and knead briefly to form smooth, cohesive ball. Transfer dough to prepared bowl and turn to coat. Cover with plastic wrap and place in turned-off oven until dough has doubled in size, about 45 minutes.
SHAPE ROLLS Punch down dough on lightly floured work surface. Divide dough into quarters and cut each quarter into 3 equal pieces. On clean surface, form each piece into rough ball by pinching and pulling the edges under so that the top is smooth. Then cup each round with your palm and move in a circular motion on an unfloured surface to shape a smooth ball. Transfer to prepared baking sheet. Cover loosely with plastic wrap and return to turned-off oven until doubled in size, about 20 minutes. (Or refrigerate rolls for up to 24 hours.).
BRUSH AND BAKE Remove rolls from oven and discard plastic wrap. Heat oven to 400 degrees. Brush rolls with egg and sprinkle evenly with ½ teaspoon salt. Bake until golden brown, about 15 minutes, rotating sheet halfway through cooking. Cool rolls on sheet 10 minutes. Serve.
MAKE AHEAD: Bake as directed until rolls are just beginning to brown and outside is set, about 5 minutes. Cool completely and freeze in zipper-lock bag. When ready to reheat, defrost at room temperature for about 1 hour and bake 10 minutes in 400-degree oven.
TO MAKE WITHOUT A MIXER: In step 2, whisk water, oil, sugar, and yeast in liquid measuring cup. Mix flour, potato flakes, and salt in large bowl. Make well in flour, then add water mixture to well. Stir until dough becomes shaggy and difficult to stir. Turn out onto floured work surface and knead until dough is smooth and forms cohesive ball, about 10 minutes. Proceed as directed.
Nifty Make-Ahead Trick: Steal a technique from commercial bakeries: par-bake. When the rolls are just beginning to brown (about 5 minutes), remove them from the oven, cool, and freeze for up to a month. To finish, let the parbaked rolls sit on a parchment-lined baking sheet at room temperature for 1 hour. Bake in a 400-degree oven until they’re golden brown, about 10 minutes.