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To make the dough, combine the flour, yeast, salt, and sugar in the bowl of an electric mixer fitted with the dough hook. Mix on low just to combine. Stir the butter into the warm mashed potatoes and stir until it is melted and mixed well. Add the water, the potato mixture, and the eggs to the flour mixture and mix on low speed for 4 minutes until all the ingredients are well incorporated. Scrape down the bowl and scrape the dough away from the hook. Raise the speed to medium and mix for 3 minutes more, kneading the dough with the dough hook a bit. The dough may be sticky and stick to the bowl, which is fine.
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Remove the dough from the bowl and spray or oil the bowl, then put the dough back in and cover with plastic wrap. Let the dough rise at room temperature for 1 hour. Transfer the dough to the refrigerator and let rise overnight (if making all in one day, let it rise for about 2-3 hours total at room temperature, until well doubled). The day you want to bake the rolls, bring the dough to room temperature for about 30 minutes, then divide the dough into even pieces 24 pieces. On a lightly floured surface, roll each piece of dough into a ball.
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Preheat the oven to 375° F. Grease cake or pie pans with some melted butter. Place the balls of dough in the pan leaving plenty of room between for rising and oven spring. In a typical glass pie plate, I can fit about 7, in a larger pie plate, I can fit 9 or so. If you end up with an odd number, you can tuck them in here and there, or bake them in muffin tins or individual custard cups. Brush the rolls with the egg wash, and save the rest of the egg wash for a second brush right before baking. Let rise for about an hour.
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When the rolls are done rising, they will appear puffed, but they do not need to be touching.
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Brush them again with the egg wash and then bake until deeply golden brown, 25 to 30 minutes. A thermometer inserted into the center should read above 200F to make sure they are done.