In a large bowl, whisk together the flour and salt. Add the cold, cubed butter. Using a pastry blender or your fingertips, cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining.
Gradually add the ice water, one tablespoon at a time, mixing gently after each addition. Stop adding water when the dough just comes together and is not too sticky.
Form the dough into a disc, wrap it tightly in plastic wrap, and chill in the refrigerator for at least 30 minutes.
In a large bowl, combine the quartered figs, chopped walnuts, brown sugar, flour, cinnamon, nutmeg, and lemon juice. Toss gently to combine everything.
Preheat your oven to 375°F (190°C). On a lightly floured surface, roll out the chilled dough into a 12-inch circle.
Carefully transfer the rolled-out dough to a baking sheet lined with parchment paper. Arrange the fig mixture in the center of the dough, leaving a 2-inch border around the edge.
Gently fold the edges of the dough over the filling, pleating as you go.
Brush the folded edges of the crust with the beaten egg wash. Sprinkle the coarse sugar over the egg-washed crust.
Bake in the preheated oven for 35-40 minutes, or until the crust is golden brown and the filling is bubbly.
Let the Fig and Walnut Galette cool on the baking sheet for at least 15 minutes before slicing and serving.