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Special equipment: a deep-fry thermometer; a flexible bench scraper
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For the dough: Heat the milk in a small saucepan over medium heat just until bubbles form around the edge.
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Transfer the milk to a small bowl and let it cool to about 110 degrees F.
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Meanwhile, in a large bowl, whisk together the flour, granulated sugar, yeast and salt.
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Add the egg to the milk and whisk to combine.
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With a wooden spoon, add the milk mixture to the flour mixture and stir it into a ball.
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Tip the dough onto a lightly floured surface and knead the dough until it is smooth, about 5 minutes.
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You can add a little bit more flour at this point if necessary.
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Next, knead the butter into the dough piece by piece, using a flexible bench scraper to scrape the butter back into the dough as you knead.
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The dough will seem sticky and buttery, but don't add more flour.
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Just keep kneading and stretching the dough until the butter is completely absorbed and the dough is smooth.
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Pop the dough into a buttered bowl, cover tightly with plastic wrap and refrigerate overnight.
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For the apple mixture: The next day, toss the apples with the cider vinegar in a medium bowl.
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In a large skillet, melt the butter over medium heat.
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Sprinkle the granulated sugar over the melted butter and cook, stirring, until the sugar turns deep golden brown, 2 to 3 minutes.
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(The butter and sugar may separate.
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Don't worry.)
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Add the apples and toss to combine.
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Cook until the apples release some of their juice and are just tender but still hold their shape, 1 to 2 minutes.
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Use a slotted spoon to transfer the apples to a plate.
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Cook the remaining liquid down until thick and syrupy, about 3 minutes more.
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Drizzle the caramel over the apples and toss gently to combine.
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Let cool to room temperature.
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Pull the dough out of the fridge.
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On a lightly floured surface, roll the dough out into a large rectangle, with the short sides at the bottom and the top, about 1/4 inch thick.
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Use a slotted spoon to transfer half the apples to the bottom half of the rectangle.
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Leave most of the apple liquid behind.
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Fold the top half of the dough over the bottom half.
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Press the dough down to seal in the apples.
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Spread the rest of the apples, without too much of that liquid, on the right half of the dough square.
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Then fold the left half of the dough over the apples and press to seal.
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Roll this into a ball, transfer it to a buttered bowl, cover lightly and set aside to double, about an hour.
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On a floured surface, tip the doubled dough out and press it into a big square.
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Use the bench scraper to cut the dough into 12 equal pieces.
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Shape each piece into a rough ball as best you can.
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Don't worry if they seem messy or apples pop out.
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Just smoosh them back together and proceed.
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Transfer the pieces to well-floured baking sheets.
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Cover lightly with plastic and let stand until puffed, about an hour.
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Heat 3 inches of oil in a large, heavy-bottomed pot to 360 degrees F. Line a baking sheet with paper towel.
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For the glaze: Meanwhile, whisk together the confectioners' sugar, milk and vanilla in a bowl.
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Add more milk or a little water if necessary.
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It should be runny.
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Very carefully transfer a puffed piece of dough to a large spider or spatula and lower it into the oil.
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Add a few more dough balls, but don't crowd the pan.
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Cook until they are golden brown and puffed, 2 to 3 minutes, flipping halfway through.
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Make sure to keep the oil temperature between 350 to 360 degrees F. Lift the fritters out with the spider and set them on the paper towel-lined baking sheet for about 3 minutes.
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Transfer the fritters to a rack and use a pastry brush to cover them in glaze.
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Serve hot or at room temperature.
