Make the pastry. Put the flour, salt and sugar into your food processor, pulse once, then add the cold, cubed butter all at once. Pulse in little quick bursts (in order to avoid warming up the butter) until everything looks uniformly like coarse cornmeal.
When the mixture has become cornmeal-like, start adding in the cold water. Begin with 3 tablespoons of cold water, and pulse quickly once or twice. What you are looking for is the moment when the whole suddenly transforms from a floury mix to a ball of dough. You will likely need to add at least one or two more tablespoons of cold water to achieve this (the amount will vary depending on the moisture content of your flour and butter).
When the dough comes together into a ball, unplug the food processor, pour the dough onto a very lightly floured surface, press it into a disc shape (I usually go with about 7 or 8 inches in diameter), wrap it in plastic and put it in the fridge for at least 20 minutes (you can leave it there for hours if you need to do the assembly and baking later).
Prepare the fruit. While the dough is chilling, prepare the apples. Slice them into a medium bowl and immediately toss with 1 tablespoon lemon juice. Then add the brown sugar (sprinkle it over the apples using your hands, breaking up any lumps), the cornstarch, nutmeg, lemon zest and the half teaspoon of salt. Toss all together with your hands. Taste one of the slices—you may want to adjust the sweetness or acidity a bit.
Put it all together. Prepare a baking sheet with a silicone baking mat or parchment paper and heat the oven to 400º.
Lightly flour your work surface. Take the dough out of the fridge, unwrap it, and roll it out with a rolling pin. You want to create a circle about 14 or 15 inches in diameter. Don't worry if it is blobby looking; you don't need to prune the edges to make it a perfect round. Transfer the dough to the baking sheet—gently peel it up from your work surface with a pastry scraper or the thin edge of a flat straight spatula, then ease it onto the rolling pin and move it over to the lined baking sheet.
Give the prepared apple slices a stir, to distribute any juices that may have exuded, then heap everything in the center of the dough. Make a nice hill. You want to ensure that you are leaving a generous ring of dough all around.
Next, fold the dough up to clasp the pile of fruit, pleating it here and there to encase everything. If you intend to serve the galette from a particular platter, make sure the footprint is right—about half the time I forget and create a slightly too wide galette which we serve straight from the baking sheet, which TBH is not only not a tragedy but is in keeping with the casual I meant to do that essence of galettes.
If the dough tears, just pinch it together to repair it. If the juices start to leak, don't worry—galettes often leak, it is part of their thing.
Finally, brush the dough with the beaten egg—you'll only use maybe half of it. Then sprinkle raw sugar lightly over the crust, dot the apples here and there with that last bit of butter, and put the whole thing in the hot oven. Bake for about 40 minutes, until the dough is beautifully golden brown and the apples are tender.
Remove the galette from the oven and set the baking sheet on a cooling rack. If you want to move it to a serving plate, wait until it has cooled down significantly.
Serve this still warm (not hot!) or serve it at room temperature. This makes a nice dessert, which if you are feeling splashy you may want to doll up with whipped cream or ice cream, and it also is nice the next morning for breakfast with a big cup of coffee. I'm having some as I write this.