Dissolve 2 tablespoons of the yeast and 1/2 tablespoon sugar in 3/4 cup of the warm milk. Make sure it is 100°F or below so it doesn't kill the yeast. Whisk in 3/4 cup of the flour to create a smooth paste. Cover and let rise in a warm spot for 20 minutes.
Combine the remaining 1/4 cup of warm milk and 1 tablespoon yeast in the bowl of a stand mixer fitted with the paddle attachment. Allow the yeast to activate for 5 minutes. Add sugar, salt, vanilla, and egg yolks. Mix until smooth. Next, add in the flour paste from the other bowl. Mix again until smooth. Turn off the mixer and add 1/2 cup of the remaining flour and the butter. Mix on low for about 30 seconds until it becomes incorporated.
Switch to a dough hook. Add the rest of the flour, about 1/4 cup at a time. Knead the dough on medium speed between additions until the dough pulls away from the sides of the bowl and is smooth. It will be very soft and moist, but should not be sticky to the point where it can't be rolled out. Cover the bowl with plastic wrap and refrigerate for 45 minutes.
Roll out the dough on lightly floured wax paper to 1/2 inch thick. Use a 3-inch biscuit cutter to cut the donuts into circles. Re-roll scraps until you have 8 donuts.
Cover the donuts with a moist towel, and let them rise in a warm spot for about 1 hour, or until they double in size and are light and puffy.
While the doughnuts are proofing, melt the shortening in a large pot until a candy thermometer reads 350˚F. It should be about 2 inches of oil. Carefully flip the donuts from the wax paper into the oil. Make sure not to crowd the pot, frying 2-3 donuts at a time. Fry for 1 to 2 minutes per side, until light golden brown. Remove with a slotted spoon onto a sugared baking sheet and cover the donuts in sugar on each side. Repeat with the remaining donuts.
Allow the donuts to cool. Using a paring knife, cut a small hole into the donut and use a piping tip to fill the donut with 2 tablespoons of Nutella.