I start by whisking the flour and salt together in a mixing bowl, then cutting in the cold butter with my fingers or a pastry cutter until the mixture looks like coarse crumbs. I add ice water one tablespoon at a time, mixing gently until the dough just comes together without feeling sticky. I shape it into a disc, wrap it in plastic, and chill it for at least 30 minutes so it firms up nicely.
While the dough chills, I heat olive oil in a large skillet over medium heat and add the sliced onion. I let it cook slowly, stirring occasionally, until it turns soft and golden, which usually takes about 15 minutes. Near the end, I stir in the garlic, sage, and thyme so the herbs release their aroma without burning.
In the same skillet, I stir the pumpkin puree into the caramelized onion mixture along with the nutmeg, salt, and pepper. I let everything warm through for a couple of minutes so the flavors blend together. Once it's off the heat, I fold in half of the parmesan cheese so it melts slightly into the warm pumpkin mixture.
I take the chilled dough out and roll it on a lightly floured surface into a rough circle about twelve inches wide. It doesn't need to be perfect since a galette is meant to look rustic and homemade. I transfer the rolled dough onto a parchment-lined baking sheet before adding any filling.
I spread the pumpkin filling in the center of the dough, leaving about a two-inch border all around. I scatter the goat cheese and remaining parmesan over the top, then fold the edges of the dough up and over the filling, pleating it gently as I go around the circle.
I brush the folded crust with the beaten egg for a shiny, golden finish, then bake the galette at 400°F for about 35 to 40 minutes, until the crust is deeply browned and crisp. I let it cool on the pan for ten minutes before slicing so the filling has time to set.