In the bowl of a stand mixer, dissolve yeast and 2 teaspoons of sugar in warm milk. Let sit for 5-8 minutes until mixture is foamy. Whisk in remaining sugar and eggs.
Attach dough hook to stand mixture. Add half of flour mixture and salt. Mix on low speed until dough begins to come together. Add remaining flour and continue to mix until dough is shaggy. Slowly add butter and increase speed to medium. Knead for 3 minutes until mixture forms a smooth dough.
Transfer dough to a lightly oiled large bowl. Cover with plastic wrap and allow to chill in the fridge overnight. (8-12 hours)
On a floured surface, roll dough to about ½-inch thick. Using a 3 ½ or 4-inch round cutter, punch out doughnuts. Loosely cover rounds with plastic wrap. Let rest for 45-60 minutes until doughnuts are puffy.
Add vegetable shortening to a large heavy bottomed saucepot (Dutch ovens and cast iron pans work best). Set over medium heat until oil reaches 350° F.
Carefully place 2-3 doughnut rounds into hot oil. Do not overcrowd. Cook for 90 seconds to 2 minutes on each side, until golden brown. Use a slotted spoon to transfer cooked doughnuts onto a paper-towel lined baking sheet. NOTE: Oil temperature will change as you add raw doughnuts. Monitor oil temperature to make sure doughnuts do not burn.
Toss warm doughnuts in sugar. Place on wire rack to cool until you are ready to fill doughnuts.