Preheat the oven to 375°F. Lightly grease two standard doughnut pans., In a medium-sized mixing bowl, beat together the butter, vegetable oil, and sugars until smooth., Add the eggs, beating to combine., Stir in the baking powder, baking soda, nutmeg, salt, and vanilla., Stir the flour into the butter mixture alternately with the milk, beginning and ending with the flour and making sure everything is thoroughly combined. The batter will be fairly thick; when you draw your spatula through the batter, it will leave a furrow., Spoon the batter into the lightly greased doughnut pans, filling the wells to about 1/4" shy of the rim., Bake the doughnuts for 10 to 12 minutes. Remove them from the oven, and wait 5 to 7 minutes before turning them out of the pans onto a rack., For cinnamon doughnuts, shake warm doughnuts in a plastic bag with about 1/4 to 1/3 cup (50g to 68g) cinnamon sugar. For sugar-coated doughnuts, shake doughnuts in a plastic bag with about 1/2 cup (57g) confectioners' sugar., For frosted doughnuts, see our three easy doughnut glazes. Sprinkle the glazed doughnuts with toasted coconut or chopped nuts., Store doughnuts, loosely covered, at room temperature for a couple of days; freeze for longer storage.