Make the pie crust: To the bowl of a food processor, add the flour, sugar, and salt and pulse to incorporate. Add half of the butter and pulse until just incorporated, then add the remaining butter and continue pulsing until the mixture looks like coarse sand. Slowly drizzle in ¼ cup of ice water while pulsing just until a shaggy dough comes together. Lightly flour a clean work surface and turn the dough out onto it. Press gently together into a ball, then flatten into an even, 1-inch thick disk. Wrap tightly in plastic and refrigerate for at least 6 hours and up to 3 days.
When you are ready to bake, light a wood fire: Use several large chunks of wood and some kindling to start the fire. Let the fire burn very hot until the flames subside and your wood has a nice base of embers beneath the actively burning wood. Nestle a large Dutch oven into the center of the embers and cover with a lid. Preheat for 30 minutes while you assemble the galette.
Make the filling: In a large bowl, toss together the apples, brown sugar, and cinnamon, then set aside.
Line a large rimmed baking sheet with aluminum foil and set aside. Lightly flour a clean work surface. Retrieve the dough, unwrap, and using a rolling pin, roll to an even circle, about ⅛-inch thick. Transfer the circle to the lined baking sheet, then add the filling to the center, leaving a 2½-inch border all around. Close the galette by slowly crimping the edges inwards while rotating the sheet pan, taking care not to cover the apples completely. Brush the surface of the dough with egg wash and sprinkle generously with sugar.
Using an infrared thermometer, check the temperature of the Dutch oven (you’re going for 375–400°F). Carefully transfer the pot to a heatproof surface, then use the foil to slide the galette gently into the Dutch oven. Cover with a lid, then return the pot to the fire and bake until the bottom and sides are browned and caramelized, 10–15 minutes.
Flip the galette: Remove the pot from the fire and remove the lid. Use the foil to lift the galette out of the pot and onto a heatproof plate. Place a second piece of foil over the top of the pastry, then invert onto a second plate. Slide the pastry, open-side-down, back into the Dutch oven, cover, and continue baking until the open side is caramelized and bubbling, 5–10 minutes more. Remove the pot from the fire, uncover, and cool slightly. Using oven mitts, carefully invert the Dutch oven onto a baking sheet, and then transfer the galette to a wire rack to cool slightly. Serve the galette warm with vanilla ice cream, if desired.