DOUGH. Beat the eggs in a large bowl. Mix in the sugar and softened shortening, then add the buttermilk. Stir in the flour, baking powder, baking soda, salt, nutmeg, and cinnamon just until a soft dough forms. Avoid over-mixing.
Divide the dough in half and place each half on a lightly floured surface. Lightly flour the tops, then roll the dough to about ⅓ inch thick. Let the rolled dough rest for 20 minutes.
Pour enough vegetable oil into a heavy, high-sided pot for the donuts to float while frying. Heat the oil to the tested frying temperature and use a deep-fry thermometer to keep it as steady as possible.
Use a donut cutter or a large round cutter to cut the donuts, then use a smaller cutter for the centers. Brush off any excess flour before frying so it does not burn in the oil.
Fry one donut first to check the oil temperature, browning, and cooking time before adding the rest. Fry 3 to 4 donuts at a time without crowding the pan. Turn as needed and fry until golden brown and cooked through. Let the oil return to the correct temperature between batches.
Transfer the donuts to a wire rack set over a baking sheet to drain and cool slightly. Fry the donut holes in the same way, reducing the cooking time as needed.
Whisk the melted butter, powdered sugar, vanilla, and 4 tablespoons half-and-half until smooth. Add more half-and-half a little at a time until the glaze is about the consistency of honey.
Dip the warm donuts into the glaze, then return them to the wire rack and let the glaze set before serving.