Preheat the oven to 400°F (200°C). Place a piece of parchment paper on a baking sheet.

In a large skillet over medium heat, melt butter and olive oil. Add sliced onions and cook, stirring occasionally, until caramelized, about 15–20 minutes.

Add sliced mushrooms to the onions. Sauté another 5–7 minutes until mushrooms release moisture and begin to brown. Stir in minced garlic, thyme, salt, and pepper; cook for 1–2 more minutes. Remove from heat and let mixture cool slightly.

On the parchment-lined baking sheet, unroll the pie crust. Sprinkle the cheese in the center, leaving a 2-inch border around the edges.

Spoon the mushroom-onion mixture over the cheese, spreading evenly but keeping the outer border clear.

Fold the crust edges over the filling, creating pleats as you go, until a circular galette forms with the center exposed.

In a small bowl, beat the egg with water. Brush the crust edges with egg wash for a glossy finish.

Bake for 25–30 minutes, or until the crust is golden brown and crisp. Let the galette cool slightly before slicing.