Make the dough. Dissolve yeast in lukewarm water. Set aside.
Heat milk in the microwave for 2-3 minutes.
While milk is heating, place shortening in a large bowl. Pour hot milk over it and whisk until mixed (there will still be a few shortening clumps).
Add sugar, salt, cinnamon, and nutmeg and mix well with a wooden spoon or the mixer whisk attachment. Add eggs and continue to mix, then stir in the yeast mixture.
Add 2 cups of flour and mix with a wooden spoon. Continue to add flour 1-2 cups at a time until the batter is no longer runny and is smooth.
Place dough on a clean, lightly floured surface and gently knead dough until the surface is not too sticky to the touch. The dough should be very soft, but not so sticky that you cannot handle it fairly easily. Place the soft dough into a lightly greased bowl. Cover the bowl and let rise for 1-1½ hours at room temperature. The dough should be about double in size when it's ready.
Place dough on a lightly floured surface and roll out to a 1-inch thickness. Use a 3½-inch donut cutter to cut donuts out, cutting as closely as possible together and saving the middles for doughnut holes. Move donuts and holes to a clean surface, cover, and let rise at room temperature for 1-1½ hours.
Make the glaze: Mix all ingredients in a large bowl. Set aside.
Once the donuts and holes are done rising, fill a large frying pan with 1½ inches of vegetable oil and heat to 365-380°F. Working in batches, fry the donuts and holes until golden brown, 2-3 minutes per side for donuts and about 2 minutes per side for donut holes.
Remove each donut from the oil and immediately transfer to the bowl with the glaze. Turn to make sure both sides are coated, then place on a wire rack over parchment paper to let the excess glaze drain. Repeat to make the remaining donuts.