Heat the butter in a skillet over low heat. Once melted, stir in the shallots with the salt. Cook on low heat, stirring occasionally, until the shallots are caramely in color, about 35 to 45 minutes. Stir in the brown sugar and cook for 10 more minutes. Remove from the heat and set aside. You can do this while the crust is chilling in the fridge!
Slice the sweet potatoes into paper thin rounds using a mandolin. You can truly have as many sweet potatoes as you want in the galette!
Remove one of the crusts from the fridge. (The other is good for another week or so, or it can be frozen.) Roll one of the pie crusts into a… “rustic” shape – no shape necessary really – until it is about 1/4 inch thick. Place the dough on a parchment-lined baking sheet. Add the sweet potatoes in the center, alternating layers between sweet potato rounds, caramelized shallots and crumbled goat cheese, leaving a 2-inch+ border of crust. I also like to sprinkle the layers of sweet potato with salt and pepper as I go. Once the filling is added, fold the crust over top of the squash. Brush the crust with the beaten egg wash. Bake the galette until the crust is golden, about 45 to 50 minutes.
Remove and let cool slightly before brushing it with the gremolata and serving.