For the Doughnuts:
For the Glaze:
Preheat the oven to 425 degrees Fahrenheit. Grease a standard 12-cup muffin tin.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter, oil, and sugars until smooth. Scrape down the sides of the bowl with a rubber spatula. Add the eggs and beat until combined. Stir in the baking powder, baking soda, nutmeg, cinnamon and vanilla. Scrape the bowl again. Then add the flour in 3 parts alternately with the milk ending with the flour (i.e. some flour, stir a bit, some milk, stir, flour, stir, rest of the milk, stir, rest of the flour) until well mixed.
Spoon the batter evenly into the muffin tins and bake for 15-17 minutes until pale golden on top. Transfer to a wire rack to cool slightly while you make the glaze. This recipe makes enough for two galze dips per muffin, but you could easily double for 4 dips (yum) or spread them with cream cheese frosting! We digress.
Make the glaze in a tall cup or mug for easy dipping. Mix together the melted butter, confectioner's sugar, vanilla and hot water. Whisk until smooth. Dip the crowns of each muffin into the glaze and return to the rack to cool a bit. Repeat as desired with the dipping.
Keep at room temperature in an air tight container for up to 3 days.