Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.
In a skillet over medium-high heat, heat the olive oil. Season the beef cubes with salt and pepper, then add them to the skillet. Sear the beef on all sides until browned, about 5 minutes. Remove from the skillet and let it cool.
In the same skillet, melt the butter over medium heat. Add the finely chopped mushrooms and thyme. Cook, stirring frequently, until the mushrooms have released their moisture and are browned, about 8-10 minutes. Let the mushroom mixture cool slightly.
Once cooled, mix the beef cubes with the mushroom mixture and stir in the Dijon mustard until evenly combined.
On a lightly floured surface, roll out the puff pastry to about 1/8 inch thick. Cut the pastry into 3-inch squares.
Place a spoonful of the beef and mushroom mixture in the center of each pastry square.
Fold the corners of the pastry over the filling, pinching to seal them tightly. Place the sealed bites on the prepared baking sheet.
Brush the tops of the pastry with the beaten egg for a golden finish.
Bake in the preheated oven for 15-20 minutes, or until the pastry is puffed and golden brown. Let cool for a few minutes before serving.