In the bowl of a stand mixer fitted with the paddle or dough hook attachment, combine the melted butter and the sugar until sandy and combined, about 30 seconds. Add in the eggs, one at a time, followed by the egg yolk, then the vanilla extract. Add in the baking powder, baking soda, nutmeg and salt, beating until combined. Add in one cup of flour, then stream in 1/4 cup of buttermilk. Continuing alternating the flour and buttermilk, ending with the flour. Dough will be very sticky. Remove the paddle/dough hook, cover with plastic wrap, and chill in the fridge for 1 hour.
In a large heavy bottomed sauce pot, heat the oil to about 325 degrees F. as read on a candy or deep fry thermometer. While oil heats, flour a large flat work surface and roll the dough out, flouring it liberally if needed to prevent sticking. Roll out to about 1/2" thickness. Using a doughnut cutter or biscuit cutters, cut out doughnut shapes and doughnut holes, if desired. Reroll and cut as needed.
When oil is hot, drop the doughnuts into the oil very carefully - I like using a slotted wire spoon for this. Fry the doughnuts for about 90 seconds per side, and doughnut holes for about 45 seconds per side, or until a deep golden brown. Remove from the oil and place on a paper towel-lined tray.
Meanwhile, while doughnuts fry, make your glaze. In a sauce pan, bring together the powdered sugar, corn syrup, vanilla and water, stirring until smooth. Drop the hot doughnuts into the glaze mixture and use a fork or spoon to coat them completely in the glaze. Place the glazed doughnuts onto a wire rack set over a rimmed baking sheet to set, about 20 minutes. Continue frying and glazing remaining doughnuts.
Doughnuts taste best made the same day, but you can store them airtight at room temperature for up to 2 days.