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For the chocolate doughnuts:
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Preheat oven to 350u00b0F.
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In a large mixing bowl, combine dry ingredients: including flour, brown sugar, granulated sugar, cocoa powder, baking soda, and kosher salt. Mix well.
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In a separate mixing bowl, whisk together wet ingredients: buttermilk, eggs, unsalted butter, Greek yogurt, apple cider vinegar, and vanilla extract. Mix well.
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Pour the wet ingredients into the dry ingredients and mix, but take care not to over-mix-just mix until barely incorporated. Pour batter into a large gallon-sized Ziploc bag with the very tip cut off (or use a piping bag, if you have one) and pipe the batter into doughnut molds. The batter should fill 3/4 of the mold, as these will rise when baked.
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Place in the oven and bake 18-20 minutes, or until a toothpick comes out clean when inserted into the center of a doughnut. Allow the doughnuts to rest in the mold once removing from the oven for 2-3 minutes, then remove from the mold, and transfer the doughnuts to a cooling rack and allow to cool completely. Repeat with remaining batter.
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For the chocolate glaze:
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Melt butter in a sauce pan over medium heat, about 2 minutes. Be careful not to let the butter burn, but just to let it melt. Add the cocoa powder and milk, stirring frequently, until it thickens and comes to a boil, about 2-4 minutes.
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Remove the pan from the heat and slowly stir in 2 cups of the confectioners sugar, adding the remaining 1/2 cup, bit by bit, as necessary to make the glaze thick and smooth. If the glaze gets too thick, you can add a little more milk to loosen.
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To glaze the doughnuts, I recommend setting up a station which is covered by parchment paper so you can move from the glaze to the cooling rack and add sprinkles without worrying about dripping.
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Holding the doughnut with your fingers, dip the tops of each doughnut into the warm glaze, do a little spin by twisting your wrist, and then shake to remove any excess glaze. Place the glazed doughnuts onto a cooling rack, glaze-side-up, and top with sprinkles (or other toppings if you desire). Repeat until all the doughnuts have been dipped, flipped, and topped with sprinkles.
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Allow doughnuts to set on the cooling rack for for 15-20 minutes to allow the glaze to harden.