Add flour, granulated sugar, and salt to a large mixing bowl and whisk until evenly combined.
Cut cold butter into 1/2-inch cubes, add to the flour mixture, and toss gently so every cube is coated in flour.
Use a pastry cutter or fork to cut the butter into the flour until the mixture looks like coarse crumbs with some pea-sized pieces of butter.
Slowly drizzle in ice-cold water while stirring with a fork, just until the dough comes together. Press any remaining dry spots in with your hands.
Shape the dough into a flat disc, wrap it tightly in plastic wrap, and chill in the fridge for 30 minutes.
Preheat your oven to 400°F (200°C) and move an oven rack to the center position. Line a baking sheet with parchment paper.
Peel 3 medium apples, remove the cores and seeds, and slice into thin wedges about 1/4 inch thick.
Add the apple slices to a mixing bowl with brown sugar, cinnamon, and lemon juice. Toss until each slice is coated, then set aside.
Lightly flour your counter and rolling pin, then roll the chilled dough from the center outward, giving it a quarter turn every few passes, until you have a 12-inch circle about 1/8-inch thick.
Carefully transfer the dough to the prepared baking sheet by loosely rolling it around the rolling pin and unrolling it onto the parchment.
Arrange the apple slices in the center of the dough, leaving a 2-inch border around the edges. Layer in a spiral or pile in an even layer, then scatter cubed butter over the apples.
Fold the edges of the dough up and over the apples, overlapping the folds slightly to form a rustic edge. Press the folds gently to secure them.
For the best shape, chill the assembled galette for 10 minutes before baking.
Crack 1 egg into a small bowl, add a splash of water, and beat until smooth. Brush the egg wash lightly over the folded dough edges.
Place the baking sheet in the preheated oven and bake for 35 to 40 minutes, until the crust is deep golden and the apples are tender when pierced with a knife.
If the edges brown too quickly, cover them loosely with foil. Let the galette cool for 10 minutes on the baking sheet before slicing and serving.