Mix butter and sugar until well blended. Add in egg, yogurt and the vanilla. Combine dry ingredients in a separate bowl and slowly mix into wet ingredients. Mix until dough is well blended. It should be the consistency of thick cookie dough- a tad sticky but you're still able to work with it.
Turn dough out onto a floured surface. Dust additional flour on top of dough as needed while rolling it 1/3"- 1/2" thick. The dough will rise some- not a ton- so if you want a traditional cake donut, I'd say you roll the dough 1/2" thick.
Begin heating the oil in a frying pan at medium heat.
Use round biscuit cutters to cut the donuts out of the dough. I purchased these Ateco cutters a while ago and have been amazed at how much I use them! If you don't have biscuit cutters you can get creative and use a glass and a bottle top! Let the dough rest for a few minutes after you cut them. This resting time also allows the oil to heat up.
Once you've got your donuts cut, fry them for about 1-2 minutes per side, until golden brown. Immediately place on a paper towel, to allow the donuts to cool. I like to fry the donut holes first, just to make sure the oil is hot enough. The donuts should immediately begin to sizzle once you place them in the oil. They cook fast too, so it's best to have all your donuts cut and ready before beginning to fry them.
You can dust with powdered sugar but I loved using chocolate glaze on mine. I simply dipped the donuts in and inverted them onto a cooling rack. Shake some sprinkles on top if you'd like and enjoy!