Part 1: Making the Flaky Galette Crust

1. Combine Dry Ingredients: In a large bowl, stir together the flour, sugar, and salt. In a separate small cup, combine the ice water and apple cider vinegar.
2. Cut in the Butter: Add the cold butter cubes to the flour and toss. Using your fingertips, quickly press each butter cube into a flat, thin sheet. Continue tossing the butter shards in the flour as you work.
3. Form and Chill Dough: Sprinkle 4 tbsp of the ice water mixture over the flour/butter. Gently toss until a shaggy dough forms, adding more water 1 tsp at a time if needed. Press the dough into a disk, wrap tightly in plastic, and chill for at least 2 hours.

Part 2: Assembling and Baking the Galette

1. Preheat & Prep Fillings: Preheat oven to $375°F$ ($190°C$) and line a baking sheet with parchment paper. In a small bowl, combine the softened goat cheese, chives, 1 tsp thyme, salt, and pepper. In a large bowl, toss the sliced squash, onion, olive oil, paprika, remaining 1 tsp thyme, salt, and pepper.
2. Roll Out Dough: On a lightly floured surface, roll the chilled dough into a 12-inch round. Transfer it to the prepared baking sheet.
3. Layer the Fillings: Spread the goat cheese mixture onto the center of the dough, leaving a 2-inch border. Arrange the seasoned squash and onion mixture evenly on top of the goat cheese.
4. Fold and Bake: Gently fold the 2-inch border of crust up and over the filling, creating pleated seals. Brush the crust with the beaten egg wash. Bake for 35 to 45 minutes, until the crust is deep golden brown and the squash is tender. Let rest for 5 minutes before serving.