To make the dough: In a large bowl whisk together the dry ingredients. In a small bowl, whisk together the egg, milk, melted butter, and vanilla. Add all at once to the dry ingredients to the mixer. Slowly mix in wet ingredients using the mixers dough hook. Mix until it makes a soft dough. Cover and let rest for 5 minutes.
Using a mixer along with the dough attachment, mix and knead until it makes a soft dough. Cover it for five minutes and let it rest! If you're in the market for a new mixer, you should really look at this stand mixer from Cuisinart. So many fancy bells and whistles and it comes in some classic colors that will go with any kitchen.
After the dough has rested, knead it again for another 6 minutes until it is soft. Place the dough into a large greased bowl and cover the top with wrap and let it rise for about 1.5 to 2 hours.
Turn the dough out onto a lightly floured surface. Then, roll the dough gently to about ¼ of an inch and cut the donuts using round cutters like these from Ateco. I used the 3 in. for the outer and the smallest one for the inner.
Cover the donuts loosely and let them rise again for another 30 minutes to an hour or until they have doubled in size. You can do the dishes, make the tea or prep for frying during this time.
Prepare a deep cast iron skillet or Dutch oven and add the oil. Use a candy thermometer to heat the oil to about 350. Depending on the size of your skillet, place 3-6 donuts at a time and make sure they are not crowded. Fry each side until golden brown. Turn over and repeat. This will take less than a minute so don't step away from the donuts while they are frying.
Drain on a baking rack, and in a Ziploc bag filled with a mixture of cinnamon and sugar, toss to coat the donuts.