Place all the flours in a bowl and mix in the salt. Add the cold butter and cut it in with a pastry cutter or with a fork, until you can still see bits of butter in the dough but they are evenly dispersed.
Drizzle in the cold water slowly, mixing with the fork, until the dough begins to hold together. Shape it into a ball and wrap in cling wrap. Using your palms, flatten the dough into a disc and refrigerate while you make the filling.
Make the filling:
Heat the oil and add the onions with a pinch of salt and ground black pepper. Let the onions cook over medium-low heat until they start to brown. Add the mushrooms and continue to cook a few more minutes until they soften.
Add the wine (turn off the stove when you add alcohol and then turn it back on), and continue to cook the onions and mushrooms for another 10-15 minutes until they are a rich brown. Add thyme and add more salt and ground black pepper as needed.
Assemble the galette:
Preheat the oven to 400 degrees.
Roll out the galette dough between two lightly floured sheets of parchment paper to make a 10-inch round. Peel off the top parchment paper and then transfer the parchment paper with the dough on it to a baking sheet.
If using mozzarella, sprinkle half of it in the middle, leaving a one-inch border. Pile the caramelized onions and mushrooms on top and spread them evenly over the cheese.
Fold the edges of the dough over the filling. You don't have to be perfect here. Gluten-free dough tends to look a little craggy at the edges and that's perfectly find. If it tears a bit as you work with it, just pinch it together with your fingers and proceed.
Brush some oil over the edges to give your crust some color. Place the galette in the preheated oven and bake 40 minutes or until lightly golden.
Let stand for 10 minutes, then slice and serve.
If you make this recipe, be sure to take a photo and tag me @HolyCowVegan on Instagram. I would love to see!