Step 1: Caramelize the onions. In a large skillet, melt 2 tablespoons butter or heat olive oil over medium-low heat. Add sliced onions and a pinch of salt. Cook, stirring occasionally, for 30-45 minutes, or until deeply golden brown and very soft and jammy.
Step 2: Sauté the mushrooms. While onions caramelize, in a separate pan, melt 1 tablespoon butter or heat olive oil over medium-high heat. Add sliced mushrooms and cook until golden brown and any liquid has evaporated, about 5-7 minutes. Season with salt and pepper.
Step 3: Prepare the filling. In a bowl, combine the cooled caramelized onions, sautéed mushrooms, shredded Gruyère, and shredded mozzarella. Stir in fresh thyme, salt, and black pepper. Mix well to ensure even distribution.
Step 4: Preheat the oven and prepare pastry. Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper. On a lightly floured surface or directly on the parchment, carefully unroll one sheet of thawed puff pastry.
Step 5: Cut the pastry. Lay the puff pastry on the prepared baking sheet. Using a sharp knife or pizza cutter, make diagonal cuts, about 1-inch apart, from the edges of the pastry towards the center, leaving a solid 2-3 inch strip down the middle.
Step 6: Fill the pastry. Spoon the onion and mushroom cheese mixture evenly down the center strip of the puff pastry, leaving a small border at the top and bottom.
Step 7: Braid the pastry. Fold the top and bottom edges of the pastry over the filling. Then, alternate folding one strip from the left side over the filling, then one strip from the right side, overlapping slightly to create a braided pattern. Continue until all strips are used. Gently press the ends to seal.
Step 8: Egg wash and bake. Brush the entire surface of the caramelized onion mushroom cheese puff pastry braid with the egg wash. Bake for 20-25 minutes, or until the pastry is puffed, golden brown, and the filling is bubbly and melted. Let rest 5-10 minutes before slicing and serving.