Preheat your oven to 400 degrees Fahrenheit and prepare a large (approximately 9-inch by 13-inch size) baking dish by greasing it with butter or cooking spray. Set aside.
Heat a large skillet over medium-high heat and add the olive oil and butter.
Add the onions and garlic and saute until the onions soften and become slightly translucent.
Add the mushrooms and continue to saute until the mushrooms begin to brown slightly and become nice and soft. The mushrooms will release some liquid, but continue cooking as it evaporates.
Add the thyme and salt and pepper and stir to combine.
Once the mushrooms begin to caramelize, add the red wine and stir well, scraping any caramelized bits off the bottom of the pan. Continue to cook as the wine cooks off, until there's almost no liquid left.
When the liquid is gone, add the flour slowly, stirring so the white of the flour begins to disappear and form a paste over the mushroom mixture.
Add the stock slowly, stirring well as it cooks to create a thick, creamy sauce.
Stir in a pinch of nutmeg.
Scrape the mushroom mixture into the prepared baking dish and top it with one sheet of puff pastry, tucking the edges of the puff pastry under to form a crust around the edges of the pan.
Using your fingers, shape the edges into a soft, fluted shape (press the pastry with your right index finger in between your left thumb and index finger to create this shape).
Dock the pastry topping with a fork in several places, or use a knife to make slits in the top. This will help the air escape while it's baking and keep the pastry in place as it bakes.
Whisk together the egg and water to make an egg wash and brush it all over the top of the uncooked pastry.
Top with some flaky sea salt.
Bake for about 25 to 30 minutes at 400 degrees Fahrenheit or until the pastry is golden brown.
Serve hot from the oven (let stand for about 5 minutes before serving) with a sprinkling of fresh thyme leaves on top.