To make the galette dough: pulse the flour, sugar and salt together in a food processor fitted with the blade attachment until combined. Add the butter and pulse until the mixture resembles coarse crumbs. Drizzle egg and pulse until a ball just begins to come together.
Form dough into a disk. Wrap in plastic and chill until firm, at least 2 hours.
Pre-heat oven to 400 F.
Once chilled, remove the dough on a lightly floured surface. Cover a baking sheet/tray/dish with parchment paper. Either roll the entire dough out and transfer to parchment paper or roll out sections at a time and build the galette crust directly on the parchment paper, using your hands to pinch the pieces together. Chill back in the fridge while prepping apples.
Thinly slice apples into roughly 1/8 thick slices or use a mandolin. Arrange the apples slices on top of dough by color (optional); gently fold edges of dough around apples.
Sprinkle apples with granulated sugar, 1.5 teaspoons of cinnamon and ¾ teaspoon of nutmeg.
Melt butter; off heat add vanilla and brush apples with 3 tablespoons of the mixture.
Brush crust with remaining tablespoon of vanilla-butter mixture, sprinkle with brown sugar and remaining cinnamon and nutmeg.
Bake, rotating once, until apples are soft and jammy and the crust is golden-brown, 40-50 minutes.