Combine the milk and the vinegar. Stir and let stand at room temperature for approximately 5 minutes and then stir again. This will make sour milk, which will curdle slightly.
In the bowl of a stand mixer, combine the sour milk with 1 tablespoon of the superfine sugar and the dry yeast. Mix well to dissolve the yeast.
Add 3/4 cup flour to the bowl, mix well to create a smooth paste and then cover with plastic wrap and let the flour mixture rest in a warm place for 30 minutes.
After 30 minutes, to the bowl with the flour mixture add the remaining 2 tablespoons superfine sugar, eggs, butter and vegetable shortening. Mix on low speed using the paddle attachment of your stand mixer until all is well combined. Then, switch to your dough hook attachment and add 1 1/4 cups flour, the salt and vanilla.
Mix well on low speed until the dough starts to come together and pulls away from the sides of the bowl. The dough should be smooth and somewhat sticky, but you should be able to roll it out at this point if you needed to (which you don't). In order to reach this dough consistency, you may need to add up to an additional 1/3 cup flour. Add any additional flour slowly, 1 tablespoon at a time.
Cover the dough with plastic wrap and allow to rest in a warm place for 30 minutes.
After 30 minutes have passed, punch down on your dough to remove any gas bubbles which may have formed. Then, cover your bowl once more with plastic wrap and keep the dough in your refrigerator for at least 1 hour and up to 24 hours.
When you are ready to make your doughnuts, roll out your chilled dough on a surface which has been lightly floured. Roll the dough to 1/2 inch thickness and use a doughnut cutter to form your doughnuts. We like to use 2 round cutters to make the large doughnuts which are more traditionally and typically Greek.
Transfer your doughnuts to a parchment lined baking sheet. When you have finished shaping your doughnuts, cover them loosely with plastic wrap and allow them to rest for 30 minutes at room temperature.
Heat enough vegetable oil to come up 1/2 inch in your frying pan. We like to use a small frying pan so that we can fry one doughnut at a time. When your oil is quite hot, carefully slip one doughnut into the oil. Cook over medium high heat for approximately 60 seconds per side. Using 2 forks makes flipping your doughnuts over quite easy.
Transfer your cooked doughnuts to a cooling rack. Repeat until all of your doughnuts are fried.
In a shallow bowl add the granulated sugar. Transfer your doughnuts, one at a time to the dish and shake them gently so that they get coated with the sugar. Flip your doughnut and repeat so that both sides are coated with sugar. Repeat with all of your doughnuts.
Enjoy!