Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.

Cut mushrooms into 1/4-inch-thick slices.

Heat olive oil and butter in a large skillet over medium heat. Add mushrooms and onion and cook, stirring occasionally, until mushrooms are golden and onions are soft and translucent, 6 to 8 minutes.

Stir in soy sauce, thyme, and garlic and cook until fragrant, 30 to 60 seconds. Season with salt and pepper; set aside.

Unfold puff pastry sheet onto a lightly-floured work surface. Gently pat down or roll out dough with a rolling pin into a 10x10-inch square. Using a sharp knife, score a line about 1/2 inch from all edges to create a border, being careful not to cut all the way through. Use a fork to generously prick pastry sheet all over inside the border.

Transfer puff pastry to the baking sheet, and brush the borders with beaten egg. Evenly spread Gruyere cheese inside the border, and top with mushroom mixture.