Make the pie dough: In a medium bowl, combine the flour, sugar, and salt. Mix with fork.
Add the butter and work into the flour with a fork until only pea-size lumps remain.
Add the apple cider vinegar and mix into the dough with your fingers. Gradually add the ice water and mix with your hands until the dough starts to come together. Add more water as needed if the dough is dry. The dough should not be very tacky or sticky. Shape the dough into a rough ball and cover with plastic wrap. Chill in the refrigerator for 30 minutes.
Meanwhile, make the apple filling: In a large skillet over medium heat, combine the sliced apples, lemon juice, cinnamon, and sugar. Cook, stirring continuously, until the apples are soft, about 10 minutes. Add the butter and stir to incorporate. Remove the pan from the heat and let the filling cool.
Preheat the oven to 425°F (220°C). Line a baking sheet with parchment paper.
On a lightly floured surface, roll out the dough into a round that is about ⅛-inch (3 mm) thick. Roll the dough around the rolling pin and transfer to the prepared baking sheet.
Spoon the apple filling into the center of the dough, leaving a 2-inch (5 cm) border around the edges. Fold the edges over the apples. Brush the egg wash over the crust and sprinkle coarse sugar all over the galette.
Bake the galette for 25 minutes, until the crust is golden brown.
Meanwhile, make the caramel sauce: In a small pot, combine the sugar, salt, and butter. Cook over medium heat until the sugar dissolves and turns amber in color, 2-3 minutes, swirling the pan occasionally.
Pour in the heavy cream and stir with wooden spoon. Remove from the heat and let cool slightly.
Drizzle the caramel sauce over the galette, then slice and serve.