For the caramelized onions: Heat a large Dutch oven or braiser over medium heat. Add olive oil followed by chopped onion and salt. Stir to coat, then cook until onion is soft, about 15 minutes. Reduce heat to medium-low/low and continue to cook, stirring occasionally, until onions are a deep golden color, about 1 hour. Stir in the vinegar and cook for another 15 minutes – the onions should deepen in color.
For the filling: When 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 you prepare the dough.
To make the galettes: Preheat oven to 425˚F. Roll prepared dough into a 12-inch circle on a floured surface. Transfer to a baking sheet lined with parchment paper. I like to take an 8″ plate, turn it upside down, and gently lay it over the dough, leaving an indentation to guide the inner circle. The goal is to leave about 2″ around the edges of the dough.
Use a sharp knife to cut slits every 3-4″ along the 2″ border of the dough Sprinkle over 2 ounces of the cheese and spoon the onion filling over the cheese to the edge of the inner circle.
In a small bowl, beat the eggs until foamy. Take one of the border parts of the dough and fold it over the filling. Brush with egg, then repeat with the next piece of dough, overlapping it slightly with the previous piece. Continue in this fashion around the border of the dough.
Bake the galette for 20 minutes, sprinkle with the remaining cheese, then bake for another 20 minutes, until the crust is crisp and golden and the filling is bubbly. Pull from the oven, crack some black pepper on top, and wait about 10 minutes before slicing and serving.