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Preheat the oven to 425°F. Lightly grease a standard muffin pan. Or line with 12 paper or silicone baking cups, and grease the cups with nonstick spray; this will ensure that they peel off the muffins neatly.
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In a medium-sized mixing bowl, beat together the butter, vegetable oil, and sugars untill smooth.
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Add the eggs, beating to combine.
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Stir in the baking powder, baking soda, nutmeg, salt, and vanilla.
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Add one-third of the flour to the butter mixture, stirring gently to combine. Gently stir in half the milk. Mix in another third of the flour, then the remaining milk. Stir in the remaining flour. Scrape the sides and bottom of the bowl, then mix until the batter is thoroughly combined.
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Spoon the batter evenly into the prepared pan (a 4-Tablespoon Scoop works well here), filling the cups nearly full.
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Bake the doughnut muffins for 15 to 17 minutes, or until they're a pale golden brown and a toothpick or thin knife inserted into the middle of one of the center muffins comes out clean.
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Remove them from the oven, and let them cool for a couple of minutes, or until you can handle them. While they're cooling, melt the butter for the topping (this is easily done in the microwave).
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Place the cinnamon sugar for topping in a shallow bowl. Dip the top of each muffin into the melted butter, then roll in the cinnamon sugar until the top is generously coated in sugar.
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Serve the doughnut muffins warm, or cool on a rack and wrap airtight. Store for a day or so at room temperature.
