Place warm water, yeast, and 1 tablespoon of sugar in the bowl of a stand mixer. Stir to combine. Let sit for 5-7 minutes until yeast mixture is foamy on top.
In a large bowl, whisk together remaining sugar, flour, and salt. Set aside.
In the bowl of stand mixer, add egg yolks and vanilla. Whisk until egg yolks are incorporated.
Add half of flour mixture. Mix on low-medium speed until flour is almost incorporated. Add vegetable shortening and continue to mix on low speed. Add remaining half of flour. Once flour has been incorporated, add chopped Oreo pieces. Increase speed and mix/knead for 2 minutes or so until the dough is smooth and pulls away from the sides of the bowl.
Cover dough with plastic wrap or kitchen towel. Allow dough to rest for 10 minutes. Transfer dough to a lightly floured baking sheet. Pat dough into a rectangle with a thickness of about 1/2-inch. Cover with kitchen towel and set up a proofing box in your oven.
In medium sauce pot, bring 3 cups of water to a boil. Place sauce pot on the bottom rack of oven. Place towel covered dough on the top rack of oven. Close oven door and let the dough rise for 45 minutes, until the dough also doubled in size. Do not open the oven door during this time, keep in as much heat and steam in oven as possible.
Remove dough from proof box oven and place on a lightly floured work surface. Pat dough into a rectangle about 18 x 13-inches. Use a sharp knife (or pizza cutter) to cut the bars into 3-inch squares.
Gently transfer squares to a lightly floured baking sheet. You will need two baking sheets. Arrange squares at least 1 1/2-inches apart from each other. Bring water in sauce pot back to a boil and place back on the bottom rack of oven. Keep squares uncovered and place let rest in oven for another 45 minutes until they have puffed up.
Fill a large pot with 3-inches of vegetable. Use a candy thermometer to measure the temperature. Heat oil to 350°F.
Remove dough squares from oven. Dust off any excess flour and carefully place 2 to 3 squares into the hot oil. Be sure not to crowd the pot. Set a timer and fry one side for 35 seconds, until light golden brown in color. Carefully turn over the bar and fry the other side for another 30 seconds. Once golden brown all over, remove cooked donuts from oil and let drain over a cooling rack set over paper towels.