Pour warm milk into the bowl of a stand mixer. Add yeast and one tablespoon of sugar. Whisk to combine. Allow mixture to sit for 5-7 minutes until fragrant and bubbly.
Whisk in egg and remaining sugar.
In a medium bowl, whisk together flour and salt. Add ⅓ of flour mixture to mixing bowl. Using a hook attachment, mix together for about 30 seconds on low speed.
Add another ⅓ of flour mixture. Mix on low speed for about 45 seconds until dough starts to form. With the mixer running on low, add butter and shortening one tablespoon at a time. Once all the fat has been added, add in remaining flour. Mix on low speed until dough forms.
Increase to medium speed and knead dough for 3-5 minutes until smooth. Dough should pull away from the sides of the bowl. Dough will be a little sticky. Do not be tempted to add more flour.
Transfer dough to a large greased bowl. Cover bowl with plastic wrap. Allow dough to rest for 30-45 minutes until doubled in volume. Punch down risen dough. Cover dough and let chill in the fridge for another 30 minutes until cold.
Meanwhile, prepare oil and pot. Attach candy thermometer to a large heavy bottomed pot (cast-iron or Dutch oven, preferred). Fill pot about 3-inches deep with vegetable oil or vegetable shortening. Set over medium heat until oil reaches 365°F.
Remove dough from fridge. Punch down chilled dough. Transfer to a floured work surface. Use a rolling pin to roll dough to ½-inch thickness. Make sure to roll out any air bubbles. Use floured 3-inch cookie cutters, stamp out desired letters, numbers, or shapes. Transfer donuts to small individual parchment papers.
Gather dough scrapes and knead together until smooth. Cover with plastic wrap and let dough rest for 15 minutes before re-rolling. Repeat punching out shapes.
Once oil comes to tempearture, carefully place 2-4 similar shaped donuts (with parchment paper attached) into hot oil using stainless steel fish spatula or something similar. Do not overcrowd the pot. Cook for 80-90 seconds on each side, until golden in color. The parchment paper will detach from the dough. Use kitchen tongs or chopsticks to discard parchment paper.
Drain cooked donuts on paper towel lined baking sheet. Keep an eye on thermometer and adjust heat accordingly to maintain even cooking temperature. Cooking oil should hover between 360-375°F.