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1. In a medium bowl, whisk together the first six ingredients. In a large bowl, whisk together the superfine brown rice flour, almond flour, potato starch, coconut flour, tapioca starch, baking powder, baking soda, and salt. Add the wet ingredients to the dry and whisk until the dough comes together. Knead it with your hands a couple of times. It will be soft.
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2. Dust a work surface with a generous amount of superfine brown rice flour. Turn the dough out onto your work surface and dust another bit of flour on top of the dough. Pat the dough down a bit. As the dough sits, it will start to get a bit more compact and will be easier to work with. If it feels too soft, you can refrigerate the dough for 15 minutes. Roll the dough to a 3/4-inch thickness.
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3. Dust a 2-inch cookie cutter or doughnut cutter with additional flour. Cut out circles and place them on a baking sheet lined with parchment paper. If you are using a cookie cutter, cut out 1/2-inch holes from the center of the disks. You can re-roll the scraps once to make more doughnuts.
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4. Chill the doughnuts in the refrigerator for 15 minutes. Meanwhile, fill a shallow cast-iron pan with 2 inches of vegetable oil. Heat the oil until it reads 340u00b0F (170u00b0C) on a candy thermometer or instant-read thermometer.
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5. Carefully add the doughnuts to the oil in batches; do not overcrowd the pan. The doughnuts will sink and then rise when they are almost done. Cook to golden brown, turning the doughnuts once. Transfer to a baking sheet lined with paper towels. Repeat the frying process with the doughnut holes. Let them cool slightly.
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In a medium bowl, whisk together all the glaze ingredients. Dip the doughnuts in the glaze and cover with the shredded coconut.
