Dissolve yeast in the 1/4 cup warm water. Add a teaspoon of sugar and stir. Let it sit for 10 minutes until foamy. This will mean your yeast is activated.
In a saucepan, add in the milk, shortening, sugar and salt. Turn on heat to medium/high. Let it come to almost a bubble. Once the milk starts to form bubbles on the outer edge, remove from the heat and let cool to luke warm.
Add the milk mixture, yeast, mashed potatoes, and beaten eggs into a large mixing bowl. Stir to combine.
Add in the 5 cups of flour and stir until most of the flour is incorporated. Remove the dough from the bowl onto a clean work surface and work in the rest of the dough. Knead the dough until smooth. It will be a soft dough.
Place 1 tablespoon of oil into a bowl. Add the dough and roll it into the oil to coat it. Cover with plastic wrap and place in a warm area until doubled. It will take about 1 hour and 30 minutes.
Lightly flour a clean surface. Turn the dough out onto the the floured surface and roll out to about 1/2 inch thickness. Cut with 3 or 4 inch donut cutter.
Place the cut out donuts onto two cookie sheets that have been greased with non-stick cooking spray. I normally lay a piece of parchment paper down onto the cookie sheet and then spray that with non-stick cooking spray. The parchment paper isn't necessary.
Cover the donuts with the plastic wrap and let it rise until doubled. This will take about 30 minutes.
While the donuts are rising for the second time, add the 3 cups of oil into a large saucepan and let it heat to 350 degrees.
Once the donuts are doubled and the oil is up to temperature, add 1 to 2 donuts into the oil. Let the donuts fry for about 2 minutes until golden brown. Flip and let fry for another minute to two minutes until that side is golden brown.
Remove from the oil and place on a wire rack to drain.
Place the granulated sugar and cinnamon in a storage bag or paper bag. Toss the hot donuts in with the cinnamon and sugar.