In the bowl of a stand mixer, dissolve yeast and 1 teaspoon 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 mixer. 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, one tablespoon at a time, and increase speed to medium. Knead for 3 minutes until mixture forms a smooth dough. Dough will be sticky, but should easily scrape from the sides of the bowl.
Transfer dough to a lightly greased large bowl. Cover with plastic wrap and allow to chill in the fridge overnight (8-12 hours).
On a floured surface, roll cold dough to about ½-inch thick. Using a 4-inch round cutter, punch out doughnuts. Loosely cover rounds with plastic wrap. Let rest at room temperature for 45-60 minutes until doughnuts are puffy.
Meanwhile, add vegetable shortening to a large heavy bottomed saucepot (Dutch ovens and cast irons pans work best*). Set over medium heat until oil reaches 350°F.
Carefully place 2 to 3 doughnut rounds into hot oil. Do not overcrowd the pot. 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 using a candy thermometer to make sure doughnuts do not burn.*
Toss warm doughnuts in graham cracker sugar. Place on wire rack to cool until you are ready to fill doughnuts.