Melt the butter on half power in the microwave or in a small pot set over low heat. Set aside to cool slightly while you proceed.
In a large bowl, stir together the flour, sugar, baking powder, baking soda, and nutmeg.
Into the center of the bowl, crack the eggs and pour the buttermilk, melted butter, and vanilla.
Using a fork, beat the eggs a bit and then gently incorporate the dry ingredients into the wet to form a rough dough. Don’t overmix or donuts will be tough.
Sprinkle a little flour on your work surface. (I sometimes like to roll out dough on a nice big piece of parchment — flour the parchment if you go that route.)
Place the dough on the work surface and knead just a few times to bring it all together. Lightly flour the top of the dough and a rolling pin, and roll out to about ¼-inch thickness.
Using a 3 ½-inch donut cutter with a 1-inch hole (or similarly sized cookie cutters, biscuit cutters or even an overturned water glass and shot glass) cut out as many donuts and holes as you can.
Gather and re-roll the dough one time to make more donuts.
In a Dutch oven or similar heavy pot with high sides, heat the oil over medium-high heat until it reaches 350°F. During cooking, you will adjust the heat as necessary to maintain the temperature as best as possible.
Fry donuts in batches, about four at a time depending on the size of your pot. It should take about 1 ½ to 2 minutes per side until donuts are golden brown and cooked through. Some cracking on the top before you flip each piece is a characteristic of this style of donut.
Cool donuts on a wire rack set over a baking sheet lined with paper towels.
Into a wide, shallow bowl, sift together the powdered sugar and nutmeg.
Pour in the milk and vanilla and stir until a nice smooth glaze forms.
Dip one face of each full-sized donut into the glaze and return to rack. You can choose whether to dip the flatter face (as shown here) or the craggy face, which will capture a bit more glaze.
When you've dipped all of the full-size donuts, add the donut holes to the bowl all at once and gently move them around with a spoon to coat with the remaining glaze, then return to rack. The glaze will dry within a few minutes.