Preheat the oven to 200 degrees, when it reaches that temperature turn it OFF (you want your oven to be warm but not hot).
In a 2 cup glass measuring cup, add the butter, milk, and water. Heat in the microwave until the mixture is warm and the butter is mostly melted (about 1 minute - 1 minute 30 seconds on HIGH). It should be at or above 115 degrees on an instant read thermometer. Make sure the butter is fully melted by stirring every minute.
In a large bowl or the bowl of a stand mixer fitted with the paddle attachment, combine the flour, yeast, sugar, and salt. Mix on low for 15 seconds. Once the warm liquid mixture reaches 110 degrees on an instant read thermometer, add along with the mashed potatoes and eggs.
With mixing speed on low, mix until a soft dough forms, then switch to the dough hook and set on the lowest mixing setting for 4 minutes. The dough will seem very tacky.
Transfer to a large, clean bowl. Cover the dough with plastic wrap or a kitchen towel and place into the warm oven to rise for about 45 minutes or until doubled in size. Just be sure to check on it at the 30 minute mark to monitor the progress.
Once the dough has doubled in size, remove from the oven and punch it down. Turn it onto a lightly floured surface and cut it into 15 equal pieces using a bench scraper or butter knife. You can use a kitchen scale to ensure accuracy if desired.
Roll into circles (see video in post) and place into a well-buttered 9x13 metal baking pan. Dust with excess flour to prevent the plastic wrap or towel from sticking. Cover with plastic wrap or a clean towel and return to the oven to rise again. Allow rolls to double in size, about 45 minutes. The sides should be touching.
Remove the rolls from the oven, and turn ON your oven to 350 degrees.
Bake for 20-25 minutes or until deep golden brown in color.
Remove and let cool for 10 minutes before pulling apart.