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To make the crust: In a large bowl, whisk together the flours, sugar, and salt. Add the butter to the bowl, then use your hands to separate the cubes and coat each one in flour. Flatten the cubes between your thumbs and forefingers. Continue to pinch the pieces and toss them in the flour until you’ve broken down the butter into pieces that range in size from lentils to almonds.
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Add the sour cream and use a fork to evenly distribute it into the flour mixture. Stir until large clumps of dough begin to form, then use your hands to gently knead and squeeze the dough together until you have a cohesive mass with only some dry bits remaining on the bottom of the bowl.
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Transfer the dough to a sheet of plastic wrap or reusable wrap and transfer any dry bits from the bowl to the top of the dough, pressing to incorporate them. Wrap the dough and use the plastic wrap/reusable wrapper to form the dough into a smooth, crack-free disk about 2" thick. Roll the disk on its edges to smooth out the sides; this will make it easier to roll your dough into a circle later. Transfer the dough to the refrigerator until firm, at least 90 minutes and up to 48 hours. While the dough chills, make the apple butter.
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To make the apple butter: In a medium heavy-bottomed saucepan, combine the applesauce, brown sugar, cinnamon, and salt. Bring to a simmer over medium heat, then cook, stirring frequently and adjusting the heat to prevent scorching, until the applesauce has reduced, darkened, and begun to stick to the bottom of the pan, about 13 to 18 minutes. Remove from the heat, stir in the boiled cider, then transfer to a liquid measuring cup or small bowl to cool completely. Your mixture should have reduced to about 3/4 cup (200g).
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To make the apple filling: In a large bowl, toss the sliced apples with the brown sugar, lemon juice, and salt.
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Preheat the oven to 425°F with a rack in the middle.
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To assemble the galette: Unwrap the chilled dough and place it on a piece of floured parchment, then let it soften for 5 to 10 minutes at room temperature.
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Sprinkle the top of the dough lightly with flour. Use a rolling pin to bang your disk into an even round about 1" thick. Rotate and flip your dough as you work, and roll the dough into a 12" to 13" circle about 1/8" thick. Add more flour as needed to prevent the dough from sticking. If your dough begins to crack or feels dry, lightly spritz or flick it with cold water. If your dough becomes delicate and warm at any point, transfer it to the refrigerator for about 5 to 10 minutes to firm up before continuing to roll.
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Once the dough is fully rolled out, spread the apple butter in a thin, even layer over the dough, leaving a small 1/2" bare border around the edges.
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Starting about 2 1/2" to 3" in from the edge of the dough, shingle the apple slices in 2 or 3 concentric circles, working from the outside toward the center. (Discard any excess lemon juice left in the bowl.)
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Use kitchen scissors or a sharp knife to cut slits from the edge of the dough to the start of the apple slices, in 3" to 4" intervals around the circumference of the galette. Fold the flaps towards the center of the galette; they’ll overlap to create a decorative border. Press lightly to seal.
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Slide the parchment and galette onto a baking sheet, then chill in the refrigerator or freezer while the oven preheats (at least 15 minutes).
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When ready to bake, distribute the cubes of butter over the apples. Brush the crust generously with the egg wash, then sprinkle the tart all over with coarse sparkling sugar.
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Bake the galette for 35 to 40 minutes, until the crust is deep brown and the apples are tender.
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Let cool on the baking sheet, then use the parchment to transfer it to a cutting board to slice.
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Storage information: Store the apple galette, well wrapped, at room temperature for 2 days, or in the refrigerator for 5 days. Reheat, wrapped in foil, in a warm oven.
