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Preheat the oven to 350 degrees F. Lightly grease a doughnut pan and set off to the side.
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In a medium bowl, mix together flour, baking powder, baking soda, salt, and chia seeds.
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In a small saucepan, heat the butter over medium fire. It will start to crackle and foam. Once the crackling stops, it will start to brown. Keep an eye on the butter and when it smells slightly nutty and looks golden, about 2 minutes, remove from the heat and transfer to a small bowl to cool slightly.
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In a small bowl whisk together egg, lemon extract, buttermilk, and lemon zest. Measure out two tablespoons of brown butter and whisk that into the bowl.
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Add the wet ingredients into the dry. Mix until combined. Do not over mix or the doughnuts will end up hard and rubbery. We want them light!
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Transfer dough to a piping bag or a large plastic storage bag with one corner snipped off. Or you can just spoon the batter into the baking pan. Fill each doughnut cavity ¾ of the way with batter. Bake for 10 minutes.
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While the doughnuts bake, make the lemon glaze. Whisk together the powdered sugar, lemon juice, lemon extract, and milk in a small bowl.
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After 10 minutes, remove doughnuts from the oven. Let them cool for a few minutes before inverting them on to a wire rack and allowing them to completely cool before glazing them.
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Dip each doughnut into the glaze about ¼ of the way down, invert, and then place on a wire rack to set.
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Doughnuts are best the same day but good up for up to 2 days.