Add the butter and oil to a large, heavy-bottomed pot with a lid over medium heat (I use a 4-quart braising pan). Once the butter is melted, add the sliced onions, stirring to coat. Cover the pot. Cook 12-15 minutes, until softened and fragrant. The onions shouldn’t require too much attention at this point, but feel free to uncover and stir occasionally.
Once softened, uncover the pot and reduce heat to medium-low. Season the onions with 2 teaspoons kosher salt, stirring to combine. Continue to cook, stirring occasionally, until tender, fragrant, and deeply browned, 30-45 minutes more. See the Recipe Notes, below, for a few more helpful pointers.
Once the onions are caramelized, stir in the fresh thyme and deglaze the skillet. Slowly pour in the dry sherry, using a wooden spoon to scrape up any browned bits formed on the bottom of the pan. Increase the heat to medium-high to bring the wine to a simmer, then simmer 2-3 minutes or until reduced. Remove from the heat and cool for galette assembly.
Meanwhile, as the onions caramelize, prepare the galette crust. To the bowl of a food processor, add the flour, chives, herbs, and salt. Pulse to combine well. Add the cold butter. Pulse just enough to cut the butter into the flour, aiming for pea-sized pieces of butter – the flour mixture should look similar to coarsely grated parmesan cheese. Pulse in the apple cider vinegar, then pulse in 1 tablespoon of ice water at a time, just until the mixture begins to pull from the side of the bowl – this usually takes 5-6 tablespoons of water.
Transfer the flour mixture to a lightly floured surface. Working quickly and carefully, use your hands to press the dough into a ball and flatten into a disc. Wrap in plastic and transfer to the refrigerator to chill. Dough can be prepped up to 3 days ahead of time.
Once the caramelized onions are cooled and the dough is prepped, assemble the galette. Roll out the herbed crust dough on a lightly floured sheet of parchment paper – aim for creating a 14-inch round about ⅛-inch thick. Sprinkle about ⅔ of the cheese over the dough, leaving about 1 inch around the border. Spoon the caramelized onions over the cheese. Fold the edges of the dough over the filling, creating a 1-inch crust. Transfer the parchment paper to a baking sheet and set in the refrigerator to chill while the oven preheats.
Preheat the oven to 425 degrees F, ensuring a rack is positioned in the center of the oven. Once the oven is preheated, remove the galette from the refrigerator. Generously brush the the crust with egg wash. Bake for 40-50 minutes, until the crust is deeply browned. About 10 minutes before the galette is ready, sprinkle the remaining cheese over the center and return to the oven to melt.