Sift the flour into a large bowl, then add the sugar and salt flour. Cut your butter into large cubes, then toss into the dry ingredients to coat. Dump everything out onto a clean work surface.
Using your hands, start to rub the butter between your fingertips until they resemble big shards. Continue tossing it in the flour as you go. If your hands tend to be warm, you can switch to a bench scraper to avoid your butter getting too warm.
Continue cutting the butter until the pieces are about the size of walnuts. Make a well in the centre of your flour mixture and add your ice water slowly. N.B. You might not need all of it, so add 30ml and use your hands to toss the flour with the water to start mixing the two together. Then add the 1 teaspoon of apple cider vinegar or vodka and toss again.
As it starts to come together, you can switch to more of a kneading motion, but go gently as you don't want to encourage much gluten development. Keep adding 10ml of water at a time until the dough is properly hydrated or you've used up all the water. It should be uniformly combined and hold together easily, but not totally smooth – you want the chunks of visible butter in it!
Form the dough into an even disc, wrap tightly in plastic wrap and refrigerate for at least 30 minutes, or up to 2 days.
For the frangipane
Beat the butter until very soft, then add the sugar and ground almonds. Continue mixing until everything is nicely combined.
Add the flour and egg, and mix again. Transfer to a bowl and refrigerate for 15 minutes while you make the caramelised apples.
For the apples
Preheat the oven to 220C / 425F and line a large baking tray with parchment paper.
Add the sugar, butter, vinegar, cinnamon and vanilla to a large saucepan over a low heat. Cook gently for a few minutes then add the sliced apples and reduce the heat to the lowest setting.
Swirl the pan so the apples are evenly coated and cook for 7-8 minutes. Remove the pan from the heat and let the apple cool until lukewarm and cool enough to handle.
To construct your galette
Dust a work surface lightly with flour and roll the dough to a circle about 12 inches in diameter. Carefully transfer to the baking tray.
Sprinkle a little flour into the centre of the pastry, then spread the frangipane paste liberally across the base, leaving a 3-inch border. Pile the apples on top, then fold the edges of the pastry over the filling. Place in the fridge for 15 minutes before baking.
Brush the dough with the lightly beaten egg, then scatter over the demerara sugar over the top. Bake for 35 - 40 minutes until the crust is golden brown.
Remove from the oven and leave to cool for 10 minutes before cutting and serving with any leftover caramel sauce, ice cream or cream.
Notes
If you want to make a vegan version of this, simply swap in a plant-based butter like Flora and use Oggs Aquafaba for the pastry wash rather than egg.