Dissolve yeast in the 1/2 cup warm water. Add a pinch of sugar to help activate the yeast.
While the yeast is proofing, in the bowl of a heavy duty electric mixer, mix together the hot potato water and the shortening until the shortening has melted.
Add the sugar, salt, and mashed potatoes and mix until everything is well blended. Allow the mixture to cool before adding the yeast and eggs, otherwise you will have dead yeast and cooked eggs. Not a pretty site.
Add the proofed yeast and the eggs to the above cooled mixture and mix until well blended.
Add 3 cups of flour and mix on medium speed until smooth.
With the mixer running, add 2-3 more cups flour, 1/2 cup at a time just until the dough leaves the side of the bowl. The dough should be soft, but not sticky. Mix on high speed until smooth and elastic.
Gather into a ball and place in a large oiled bowl. Turn the dough over once in the bowl so that the oiled side of dough is facing up. Place plastic wrap tightly over the bowl and put in the refrigerator for 2-24 hours. (I had my dough in the fridge for almost 24 hours.)
Remove dough from refrigerator and punch down. Knead the dough by hand for a minute or two to work out the large gas bubbles. Return the dough to the bowl, cover, and allow to warm for about 30 minutes.
Shape rolls and place in well oiled pan. Put in a warm place and allow to rise until about double in bulk. Bake in a 375-degree oven until golden brown, about 20 minutes.