Mix together 1 cup flour, the sugar, baking soda, baking powder, salt, and nutmeg in a large bowl.
Mix together butter, buttermilk, and eggs in a large measuring cup. Add the wet ingredients to the dry in the mixing bowl, and beat either using an electric mixer on medium speed or by hand until smooth. Decrease the speed to low, and add the remaining flour. Mix until just combined, making sure to scrape the edges to ensure that all the liquid is incorporated. The consistency should be somewhere between cookie dough and cake batter (moist and tacky).
Add the shortening to a large kettle and attach a candy thermometer to the side. Gradually heat the shortening over medium-high heat to 375F. While it's heating, turn the dough onto a floured work surface. Roll out with a heavily floured rolling pin until the dough is about 1/2" thick. Stamp out the dough rings with a donut cutter (or use two circle cutters with the second being about half the diameter of the larger one). Transfer the rounds to a large wire rack, gather the scraps, and repeat rolling and stamping until all the dough is used.
Carefully drop the dough rings into the hot oil four or five at a time. Use tongs to turn the donuts as they rise to the surface. Fry the donuts until they're golden brown, about 50 seconds per side. Drain on a paper towel-lined wire rack. Let the frying oil return to temperature before adding the next batch. Repeat until all the donuts are cooked.
For the glaze, mix together all the glaze ingredients in a mixing bowl until completely smooth.
Dip the cooked donuts into the glaze until about halfway submerged. Remove the donuts from the glaze, then return to the wire rack set over a baking sheet, allowing to cool on the dry side.