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