In a small bowl heat up the milk in the microwave between 100 and 110 F (37 - 43 C) Or heat it in a pot on the stovetop. Add a pinch of the sugar and also the yeast. Mix with a fork and set aside for 5 to 10 minutes until the yeast gets nice and foamy/frothy.
Now melt the butter until just barely melted and add that along with with the egg and sugar. Mix those ingredients together with a whisk.
In a separate bowl whisk together the flour, salt, and nutmeg. Then add it a third at a time to the main bowl. Use a wooden spoon to mix the ingredients until a dough forms. Save about two tablespoons of the flour mixture for later.
Dump the dough onto a work surface and knead it for 10 minutes. The dough will be sticky, instead of using more flour, use a bit of oil on your hands to knead the dough. Adding more flour will make the donuts less light and pillowy when fried.
Once the dough has been kneaded, shape it into a ball and place it into a large bowl with a bit of oil. Coat the dough with the oil and then cover the bowl with a damp cloth or plastic wrap and allow the dough to rise for 90 minutes to 2 hours until the dough more than doubles in size.
After the dough has risen, roll it out onto a lightly floured work surface, use the remaining flour from before. Make it about 1/4 of an inch thick. Then use a donut cutter or a biscuit cutter and pastry trip to cut out the doughnut shapes.
You can combine all the scraps and then roll them out or just leave the scraps as is for unique looking donut shapes.
Place the cutout donuts onto a sheet pan lined with parchment paper. And then cover loosely with plastic wrap or a cloth and let them rise for 1 hour.
20 minutes before done rising add oil to a large pot until it is about 1 inch to 1 1/2 inches deep. Set the heat to medium and heat the oil to 325 F/162 C.
Once the oil is at temp, you can fry up those donuts. Gently drop in a couple of donuts and all them to cook for about 1 minute per side until golden brown on both sides. Use a fork or spider strainer to turn them over and then remove them when done.
Place them on a wire rack to drain. Make sure to fry up those donut holes and other scraps.
Create the glaze by melting the butter in a bowl and adding the powdered sugar and enough milk to get a nice thin consistency or thick if desired. Dip the donuts in the glaze and make sure they are evenly coated. Then place them on a wire rack to dry. Once dried gobble up those homemade glazed donuts. Enjoy!