To make the doughnut holes: Preheat the oven to 350 degrees F and lightly butter (or spray) a mini muffin pan.
Sift the flour, baking powder, baking soda, salt and nutmeg into a large bowl and then whisk to ensure everything is fully incorporated. Set aside.
On medium-low speed, combine the oil and sugar in the bowl of a stand mixer, fitted with the paddle attachment.
Add the eggs, one at a time, mixing after each addition, and continue mixing until combined.
Add the dry ingredients, alternating with the buttermilk, in 3 installments, beginning and ending with the dry. Stop the mixer and finish mixing the dough with a rubber spatula.
Scoop the dough by the tablespoonful (I used my smallest ice cream scoop for precision’s sake), filling the mini muffin pan and place the pan in the middle rack of the oven.
Bake for about 10 minutes, rotating the pan halfway through the bake time. The muffins are tiny and bake quickly. They are done when a toothpick comes out clean (full disclosure: I have a convection oven and although I drop the temperature of my oven by 25 degrees – ie: i baked these muffins at 325 degrees – my oven still seems to bake things much more quickly than conventional ovens. Long story short: check your muffins after 8 minutes).
To make the coating: Melt the butter in a small bowl, deep enough to roll a doughnut hole around in. Whisk the sugar and cinnamon together in another small bowl with some depth.
Assemble: Remove the doughnut holes from the oven and, as soon as they have cooled enough to handle, begin dipping them one at a time in the bowl of butter (I used a large spoon for lowering them in to, and removing them from, the butter) and then rolling them in the sugar/cinnamon mixture.
Transfer the coated holes to a wire rack. Let cool and consume immediately (although you really don’t have to: these tasty treats last surprisingly well if tightly wrapped with plastic wrap and left on the counter for a day or two).