Preheat the oven to 425˚ F. Line a baking sheet with a silicone baking mat or parchment paper. Place the dough on a lightly floured work surface and roll out to just over ¼-inch thick. Use a 2-inch round cookie or biscuit cutter to cut out as many rounds of dough as possible. (Keep them close together to get as many as you can in your first go. Unlike cookie dough, puff pastry doesn’t take well to being rerolled a second time, as the orientation of the layers is affected and it isn’t pretty when it puffs up.) Use a smaller round cutter (about 1 inch diameter) to cut the center out of half of the dough rounds.
Place the intact dough rounds on the prepared baking sheet. Brush lightly with the egg wash. Top each with one of the dough rings and brush the top of the ring with egg wash as well. Transfer to the oven and bake for 8 minutes. Rotate the pan 180˚ and reduce the heat to 350˚ F. Bake 5-10 minutes more, until golden and puffy. Maintain the oven temperature.
Meanwhile, in a large skillet, heat the olive oil and butter over medium-high heat. Add the mushrooms to the pan and cook, stirring occasionally, until they have softened and most of their juices have evaporated. Stir in the garlic and thyme and cook just until fragrant, about 1 minute. Season with salt and pepper to taste. Add the wine to the pan and cook, scraping any browned bits from the bottom of the pan, until the wine has mostly evaporated. Remove the pan from the heat.
If the centers of the dough rounds have risen a lot, tamp them down so the center well is as deep as possible. Spoon some of the mushroom filling into each puff pastry shell. Top each with a bit of the garlic herb cheese. Return the pan to the oven 1-2 minutes to soften the cheese. Garnish with additional fresh thyme, if desired. Serve immediately.
*This same concept can certainly be achieved using frozen puff pastry dough, but I have only tried with my homemade version so I cannot comment on what would be an equivalent amount of the store-bought kind.
**If you are local to the Indy area, I highly recommend the garden herb fromage blanc from Traders Point Creamery. It is divine!