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Preheat oven to 325 degrees F. Grease or spray doughnut pan or muffin tin.
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In a large bowl, whisk together flour, sugar, baking powder and salt.
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In a liquid measuring cup, whisk together buttemilk, eggs and vanilla then stir into flour mixture.
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When mixture is almost combined (you still have floury streaks), add the melted butter and stir until incorporated.
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Spoon batter into a large zippered plastic bag. I use freezer bags because they're sturdier.
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Snip a corner off and pipe batter into doughnut pan, about 2/3 full. (You can also just spoon the batter into the pans.)
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Bake for 8-11 minutes or until firm. Doughnuts will not be golden brown on top, they'll appear light, but will spring back when touched. Mine took exactly 8 minutes. When you take them out of the pan, the flip side will be lightly golden.
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Allow to cool for 5 minutes in the pan, then remove to a rack.
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Make the glaze. Mix confectioner's sugar, water or milk, vanilla and peppermint extract in a small bowl. Add enough water to get a good drizzling consistency.
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Dip each doughnut top, warm or cool, into the glaze, scraping excess off with a spoon.
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Sprinkle with chopped white chocolate peppermint chips. Allow icing to set before packing up for storing.