Prepare the dough: In a large bowl, whisk together flour and salt. Cut in cold, diced butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces. Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together. Form into a disk, wrap in plastic, and chill for at least 30 minutes

Sauté mushrooms: Heat olive oil in a large skillet over medium-high heat. Add the sliced mushrooms (cremini, shiitake, and oyster) and cook for 8-10 minutes, stirring occasionally, until they are browned and tender and most of their liquid has evaporated. Add the minced garlic and chopped fresh thyme, and cook for another minute until fragrant. Season with 0.5 tsp salt and 0.25 tsp black pepper. Remove from heat and let the mixture cool slightly.

Assemble the galette: Preheat your oven to 400°F (200°C). On a lightly floured surface, roll out the chilled dough into a rough 12-14 inch circle. Carefully transfer the dough to a baking sheet lined with parchment paper. Spread the shredded Gruyère cheese evenly over the dough, leaving a 2-inch border around the edges. Spoon the cooled mushroom mixture over the cheese.

Fold the edges: Gently fold the exposed dough border over the filling, pleating it as you go, to create a rustic, free-form edge. In a small bowl, whisk together the beaten egg and 1 tbsp water to create an egg wash. Brush the egg wash over the folded dough edges.

Bake the galette: Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown and the cheese is bubbly and lightly browned. Let the galette cool on the baking sheet for 10-15 minutes before slicing and serving.