Combine flour, granulated sugar, cinnamon, and Kosher salt in the bowl of a food processor. Add cold, cubed, unsalted butter. Pulse. Add ice water and pulse.
Place sliced apples, brown sugar, lemon zest, fresh lemon juice, vanilla extract, cornstarch, ground cinnamon, and Kosher salt in a large bowl, then toss to thoroughly mix.
Line a rimmed baking sheet with parchment paper. Lightly flour a work surface, then use a rolling pin to roll the chilled dough out into a circle. Carefully roll the dough up around the rolling pin, then transfer it to the prepared baking sheet.
Use a slotted spoon to transfer the filling to the crust, allowing excess juice to drip off and leaving about 2-inches of dough overhang on all sides. Press down slightly to seal. Dollop the top of the galette with cubed butter.
In a small bowl, whisk together large egg and heavy cream, milk, or water, then use a silicone pastry brush to brush all over the pie dough. Sprinkle the pie dough with coarse turbinado sugar.
Bake until the crust is golden brown and the filling is bubbling. Cool to allow the filling time to set. Slice, then serve with salted caramel and vanilla ice cream.