Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper.
In a skillet over medium-high heat, melt 2 tablespoons of butter and add the chopped mushrooms. Cook until browned and liquid has evaporated, about 5-7 minutes.
Add minced garlic and thyme to the skillet and cook for an additional minute. Season with salt and pepper to taste, then remove from heat and let cool.
In a mixing bowl, combine the beef pieces with the cooled mushroom mixture, stirring in Dijon mustard until well combined.
Roll out the puff pastry sheet on a lightly floured surface to about ⅛-inch thick and cut into 3-inch squares.
Place a tablespoon of the beef mixture in the center of each pastry square.
Fold the pastry over the filling to create a triangle or rectangle, sealing the edges by pressing with fingers or using a fork. Place on the prepared baking sheet.
Brush the tops of the pastries with the beaten egg for a golden finish.
Bake in the preheated oven for 20-25 minutes or until the pastries are golden and puffed.
Remove from oven and let cool for a few minutes before serving. Enjoy warm!