Mix the flour, baking powder, salt and nutmeg in a medium bowl and set aside.
In the bowl of your stand mixer, mix the sugar and shortening for about a minute on low. Add the egg yolks and mix for another minute on medium speed. Make sure to scrape the sides of the bowl to incorporate all the ingredients.
Add 1/3 of the dry ingredients to the wet ingredients and mix on low until just combined.
Add 1/2 the Greek Yogurt and mix on low speed until just combined.
Add 1/3 of the dry ingredients and mix.
Add the last 1/2 of the Greek Yogurt and mix.
Add the last 1/3 of the dry ingredients and mix. The dough will be sticky.
Transfer dough to a clean bowl and cover with plastic wrap and refrigerate for 45 minutes.
Heat up your oil in a large pot or high-sided frying pan to 325 degrees.
Meanwhile, roll out the chilled dough on a floured surface to about 1/2" thickness. Cut out the donuts with a donut cutter or cup. (Anything round will work!) I use a water bottle cap for the smaller hole in the center.
Add donuts and donut centers to the hot oil a few at a time. Once the donuts float let them fry for 15 seconds and then gently flip them over. Fry for another minute to minute and half until golden brown and slightly cracked. Then flip and fry the first side for another minute. If the donuts are cooking too fast and getting dark brown quickly, check your oil temp.
Transfer to a cookie rack that has been placed over paper towels.
Mix glaze ingredients together until sugar is fully incorporated.
Drizzle the glaze over the donuts, or dip the top of your donuts into the glaze. Let it dry just a bit and serve!