Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a large skillet over medium heat, cook the ground beef, onion, and garlic until the beef is browned and the onion is soft, about 5-7 minutes. Drain excess fat.
Stir in the salt, pepper, thyme, Worcestershire sauce, and mushrooms. Cook for an additional 5 minutes until the mushrooms are tender. Remove from heat and let cool slightly.
In a bowl, mix the beef mixture with the cream cheese until well combined.
Roll out the puff pastry on a lightly floured surface to about 1/8 inch thick. Cut into 4-inch squares.
Place a tablespoon of the beef mixture in the center of each square. Fold the pastry over to form a triangle and seal the edges by pressing with a fork. Brush the tops with the beaten egg.
Place the turnovers on the prepared baking sheet and bake for 20-25 minutes, or until golden brown.
While the turnovers are baking, prepare the sweet chili wine sauce. In a small saucepan over medium heat, combine sweet chili sauce, red wine, and soy sauce. Bring to a simmer.
Stir in the cornstarch mixture and cook until the sauce thickens, about 2-3 minutes. Remove from heat.
Serve the turnovers warm with the sweet chili wine sauce drizzled on top.