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Preheat the oven to 350°F. Lightly grease a standard doughnut pan. (See "tips," below, for an alternative pan.)
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In a medium bowl, whisk together the flour, buttermilk powder, baking powder, salt, apple pie spice, and sugar until well blended. Stir in the diced apple. Set aside.
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In a separate small bowl or measuring cup, whisk together the melted butter, egg, boiled cider, water, and vanilla.
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Fold the wet ingredients into the dry ingredients until the batter is evenly moistened and smooth (except for the apple pieces).
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Fill the wells of the doughnut pan nearly to the rim.
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Bake the doughnuts for 12 to 14 minutes, or until a toothpick inserted into the center of one comes out clean.
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Remove the doughnuts from the oven, and loosen their edges.
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After about 5 minutes, transfer them to a rack to cool completely.
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To make the caramel: Combine the milk and caramel in a small saucepan set over medium heat or in a microwave-safe dish.
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Heat, stirring occasionally, until the caramel melts and the mixture is smooth.
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Spread the doughnuts with caramel (or dip their tops in caramel), sprinkle them with sea salt, and return them to the rack until the caramel is set.