In a mixing bowl, combine all-purpose flour, white whole wheat flour, and salt. Incorporate cold cubed unsalted butter until the mixture resembles coarse crumbs. Add the beaten egg and cold water gradually, mixing until a shaggy dough forms. Shape it into a disk, wrap in plastic, and chill in the refrigerator for at least 30 minutes.

While the dough chills, heat a skillet over medium heat and melt unsalted butter until golden brown. Stir in finely chopped fresh sage and cook for another minute. In a bowl, mix the browned butter with pumpkin puree, a large egg, and kosher salt until smooth.

Preheat your oven to 425°F (220°C). On a lightly floured surface, roll out the chilled dough into a 12-inch circle, about 1/8-inch thick. Transfer the dough to a parchment-lined baking sheet. Spread the pumpkin filling in the center, leaving a 2-inch border. Layer the caramelized onions, crumbled goat cheese, and a drizzle of honey atop the filling.

Fold the edges of the dough over the filling, creating pleats as you go. Brush the exposed dough with an egg wash made from a beaten egg. Sprinkle flaky sea salt over the crust if desired. Bake for 25-30 minutes until the crust is golden brown and the filling is set.

Once baked, remove the galette from the oven and let it cool for about 10 minutes. Cut into wedges and serve warm, showcasing the delicious flavors of the Pumpkin and Goat Cheese Galette.