Mix dry ingredients. In a bowl, whisk together flour, salt, and sugar.
Cut in the butter. Use a pastry cutter to cut the cubed butter into the dry mix until coarse and crumbly. Add ice water. Pour in the water and gently mix until the dough comes together when pressed.
Form and chill. Form a flat disk with the dough, tightly wrap it, and refrigerate for at least 1 hour.
Make the filling. Heat the oil over medium/high heat. Add the mushrooms and sauté until browned. Add the onion and reduce the heat to medium-low; season with salt and pepper. Sauté until the onions are tender, about 3 minutes. Set aside.
Prep. Preheat the oven to 400°F. Line a baking sheet with parchment paper.
Assemble. On a lightly floured surface, roll the dough into a 12-inch circle about a ¼-inch thick. Carefully transfer the disc to the prepared baking sheet.
Cheese mix. In a bowl, mix the ricotta cheese, ¼ tsp salt, and ¼ tsp pepper. Spread over the disc, leaving about a 1-inch edge unfilled, and sprinkle the shredded cheese on top.
Top with filling. Add the sautéed mushroom mixture over the cheeses.
Shape. Fold the edges of the dough over the filling, pinching overlapping edges as needed.
Bake. Bake 20-25 minutes or until the crust is golden and the filling is bubbling. Let the galette cool on a cooling rack for at least an hour before slicing.