Season the beef steaks generously with salt and pepper. Heat olive oil in a skillet over high heat and sear steaks for 1 minute on each side until browned but rare inside. Remove and let cool completely.

In the same skillet, add chopped mushrooms, shallots, garlic, and thyme. Cook over medium heat for about 10-15 minutes until all liquid has evaporated and the mixture (duxelles) is dry. Let cool.

Once the beef is cool, brush each steak on all sides with a thin layer of Dijon mustard.

Lay out plastic wrap on a counter. Place 2 slices of prosciutto slightly overlapping, spread a quarter of the mushroom mixture over the ham, then place a steak in the center. Use the plastic wrap to roll the prosciutto tightly around the beef. Repeat for all steaks and chill for 15 minutes.

Preheat oven to 400°F (200°C) and line a baking sheet with parchment paper.

Cut puff pastry sheets into 4 squares large enough to encase the steaks. Remove plastic wrap from beef and place one bundle in the center of each pastry square.

Fold the pastry corners up over the beef to seal, trimming excess dough. Place seam-side down on the baking sheet.

Brush the tops of the pastry with the beaten egg and score lightly with a knife for decoration.

Bake for 20-25 minutes until the pastry is golden brown and puffy. Let rest for 10 minutes before serving.