Make the mushroom filling: In a 12-inch cast iron or stainless steel skillet over medium-high heat, melt together 1 tablespoon of Private Selection™ Salted French Butter and the Private Selection™ Italian Extra Virgin Olive Oil. Once it starts to foam, add the mushrooms and thyme and stir to coat. Sear for 2–3 minutes without stirring, until the bottoms of the mushrooms are golden brown, before stirring again and continuing to cook without disturbing. Repeat, adding more butter as needed if the pan looks too dry, until the mushrooms are browned and cooked through, 7–8 minutes total.
Reduce the heat to medium-low. Add the garlic and pepper and cook for another minute, until the garlic is fragrant.
Remove the pan from the heat and deglaze with the sherry vinegar, stirring to scrape up any browned bits from the bottom of the pan. Remove the thyme sprigs, then set aside to cool.
Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
Roll out the puff pastry sheet if needed to a 9-inch square, then cut into 9 3-inch squares and transfer to the prepared baking sheet. Use a paring knife to lightly score a ½-inch border around the edges of each square, making sure not to cut all the way through. Use a pastry brush to brush the egg yolk mixture over the borders of each square.
Place about 2 teaspoons of Private Selection® Alpine Blend Cheese in the center of each pastry square, then top with about 1 tablespoon of the mushroom mixture, arranging the mushrooms in a nice pattern, if desired. Top each pastry with a small pinch of Private Selection® Alpine Blend Cheese.
Bake the squares for 15–17 minutes, until the pastry is puffed and golden brown. Remove from the oven and let cool slightly.
Use your fingers to strip the leaves from the thyme sprigs, then garnish the mushroom bites with the thyme leaves just before serving.