Make the Crust: In a large bowl, whisk together the flour and salt. Add the cold butter and use a pastry blender or your fingers to cut the butter into the flour until the mixture resembles coarse crumbs. Gradually add the ice water, one tablespoon at a time, mixing until the dough just comes together.

Chill the Dough: Form the dough into a disc, wrap it in plastic wrap, and refrigerate for at least 30 minutes.

Prepare the Apple Filling: In a large bowl, combine the sliced apples, sugar, flour, cinnamon, and nutmeg. Toss gently to coat the apples evenly.

Roll Out the Dough: Preheat oven to 375°F (190°C). On a lightly floured surface, roll out the chilled dough into a 12-inch circle. Transfer the dough to a baking sheet lined with parchment paper.

Assemble the Galette: Arrange the apple slices in the center of the dough, leaving a 2-inch border. Dot the apples with the butter pieces.

Fold the Edges: Carefully fold the edges of the dough over the apple filling, pleating as you go.

Egg Wash and Sprinkle: Brush the crust with the beaten egg and sprinkle with the remaining tablespoon of sugar.

Bake the Galette: Bake for 35-40 minutes, or until the crust is golden brown and the apples are tender.

Make the Salted Caramel: While the galette is baking, make the salted caramel. In a medium saucepan, melt the sugar over medium heat, stirring constantly until it is golden brown.

Add Butter and Cream: Remove the saucepan from the heat and stir in the butter until melted. Gradually pour in the heavy cream and stir until smooth. Return the saucepan to low heat and cook for 1-2 minutes, stirring constantly, until the caramel is smooth and thickened.

Add Salt: Remove from heat and stir in the sea salt. Adjust the amount of salt to your taste.

Drizzle and Serve: Let the galette cool slightly before drizzling with the salted caramel. Serve warm and enjoy!