Scrub the potato and poke several holes all over using a fork. Place the potato in the preheated oven and bake until tender, about 1 hour. Take the potato out of the oven and allow to cool for 10 minutes.
Cut the potato in half lengthwise and scoop out the white flesh into a large mixing bowl, discard the skins. Use a fork to thoroughly mash the potatoes and then set aside.
In a separate small bowl mix together the yeast and water and set aside.
Add the butter, salt, flour, and 1 of the eggs to the mashed potatoes and stir until thoroughly combined. Pour the yeast/water mixture in with the potato mixture. Use your hands to mix everything together until a dough is formed then knead the dough by hand for about 2 minutes.
Form the dough into a ball and transfer to a large clean bowl. Drizzle the dough with the olive oil and flip around a couple times to coat in the oil. Loosely cover the bowl and set in a warm place to rise until doubled in size, about 1 hour.
Preheat the oven to 375°F. Line a 9x13 baking dish with parchment paper.
Punch down the dough and separate into 12 equal size balls. Lay the dough balls in the lined baking dish about 1/2-inch apart and sprinkle the tops with just a little bit of flour. Let sit uncovered for 20 minutes.
Lightly beat the remaining egg then brush the tops of the rolls with some of the beaten egg, discarding any leftover egg.
Bake the rolls in the preheated oven until golden brown, about 25 minutes.
Sorry to hear your potato rolls didn't rise Kruzer. I've made this recipe many
times and have never had a problem. Do you think it could have been bad/old
yeast? I can't think of any other reason why it wouldn't rise.