Whisk water, oil, and sugar together in a 4-cup liquid measuring cup until sugar dissolves.
In a stand mixer fitted with dough hook, mix flour, potato flakes, yeast and 11/2 teaspoons salt on slow speed unit combined. Add water mixture in steady stream and mix until dough comes together, about 1 minute. Increase speed to medium low and knead until dough is smooth and comes away from bowl, about 6 minutes.
Turn dough onto a lightly floured counter and knead briefly to form a smooth ball. Transfer dough to a bowl slightly coated with oil and turn to coat. Cover with plastic wrap and place in a warm place until doubled in size, about 45 minutes.
On lightly floured counter, gently press down dough. Divide into quarters and cut each quarter into three pieces. Form each piece into a ball by pinching and pulling edges under so that top is smooth. Cup each ball with palm and move in circular motion to shape into a smooth taught ball.
Transfer to baking sheet lined with parchment and cover loosely with plastic that you have first sprayed with cooking spray. Put in a warm area to rise, about 20 minutes.
Remove plastic wrap and brush rolls with egg and sprinkle with salt. Bake until rolls are just beginning to brown and outside is set, about 5 to 10 minutes.
Can be refrigerated after shaped into rolls for up to 24 hours. Bake directly from refrigerator.