Combine shortening and milk in a saucepan. Heat until the shortening melts. Set aside.
Prepare instant mashed potatoes (you can also use regular mashed potatoes, even leftover ones from a previous meal). Set aside.
In a separate bowl, combine 3 cups of the flour with sugar, salt, and yeast.
Add the warm milk and shortening to the bowl with dry ingredients.
Add beaten eggs and mashed potatoes to the bowl.
Add remaining flour, one cup at a time, until you have a soft, slightly sticky dough. Be careful not to add too much.
Grease hands and knead the dough. Knead for a few minutes, making sure the dough remains soft and slightly sticky.
Allow the dough to rest for 10 minutes.
Roll dough out on a greased surface. Roll out to about 1.5" thick. Use a doughnut cutter to cut out the doughnut shapes.
Place doughnuts on baking sheets lined with parchment. Set the baking sheets aside and let rise while you cut out the rest of the dough.
Make the glaze. In a small saucepan, mix confectioner's sugar, cornstarch, and vanilla with enough hot water to make a thin glaze. You don't want it too watery. Keep warm on lowest heat setting, stirring occasionally, until you're ready to glaze the doughnuts.
Heat oil in a deep saucepan over medium high heat to 375 degrees F. Use a candy thermometer to make sure the oil is the right temperature. You can also test it by dropping a piece of the dough into the oil. If it's ready, the dough will turn brown in 30 seconds.
Carefully drop doughnuts into oil. Brown one side, then turn over quickly with tongs, and fry on the other side until golden. Don't let your oil get too hot, or the doughnuts will burn. Be careful around the hot oil, and don't leave it unattended.
Remove cooked doughnuts from the hot oil with tongs, and lay them on a plate. While they're still warm, dip them in the glaze and lay on parchment paper to dry.