For the caramelized onions: Heat a large Dutch oven or braiser over medium heat. Add the olive oil followed by the sliced onions and salt. Stir to coat, then cook until the onions soften, about 15 minutes. Reduce the heat to medium-low/low and continue to cook, stirring occasionally, until the onions are a deep golden color, about 1 hour. Stir in the vinegar and cook for another 15 minutes—the onions should continue to deepen in color.
For the filling: Once the onions are caramelized, stir in the broth, thyme, Worcestershire sauce, nutritional yeast, and shiitake powder. Cook for a few minutes, until the mixture thickens slightly. Turn off the heat, remove the thyme sprigs, and let the filling sit while preparing the dough.
To make the galette: Preheat the oven to 425˚F. Roll the prepared dough into a 12” circle on a floured surface. Transfer to a baking sheet lined with parchment paper. I like to take a 8” plate, turn it upside down, and gently lay it on the dough to leave an indentation to guide the inner circle. The goal is to leave about 2” around the edge of the dough.
Use a sharp knife to cut slits every 3-4” along the 2” dough border. Sprinkle on 2 ounces of the cheese and spoon the onion filling over the cheese, spreading it out to the edges of the inner circle.
In a small bowl whisk the egg until frothy. Take one of the border sections of dough and fold it over the filling. Brush with the egg, then repeat with the next section of dough, overlapping it slightly with the previous section. Continue in this fashion all the way around the dough border.
Bake the galette for 20 minutes, sprinkle with the remaining cheese, then bake for 20 minutes more, until the crust is crisp and golden and the filling is bubbling. Pull from the oven, crack some black pepper on top, and wait about 10 minutes to cut and serve.