This rises overnight so factor in when making. In the bowl of a stand electric mixer fitted with a dough hook, combine the flour, sugar, yeast, and salt. Mix on low speed to combine, about 30-60 seconds.
In a separate bowl, stir together the mashed potatoes and melted butter.
Add the mashed potato mixture, along with both eggs and the warm water, to the stand mixer bowl.
Mix on low speed until a dough forms, about 4 minutes.
Turn up the speed to medium, then continue to mix for another 3-4 minutes. You should expect a fairly smooth but slightly sticky dough.
Remove the bowl from the stand mixer, cover it with a kitchen towel, and let the dough rise at room temperature for an hour.
Transfer the bowl to the refrigerator to rise overnight. This dough can be made up to 2 days in advance.
Once ready to bake, remove the dough from the fridge and begin portioning (the dough is a bit more forgiving while its a little colder).
Lightly flour a large clean surface. Carefully tear the dough into 12 evenly sized pieces (weighing 50 oz). Roll each piece into a tight ball. Transfer the tightly rounded dough into a large greased sheet pan spaced a bit apart. Cover the rolls with a clean, lightly damp kitchen towel and let them rise for an additional hour. They should be visibly puffy.
Preheat the oven to 375 degrees. Place the oven racks in the middle of the oven.
Remove the towel from the rolls and bake the rolls until golden-brown and cooked through, about 20-25 minutes. Remove the rolls from the oven and immediately brush them with the herb butter. (see below). Serve hot and fresh!