Preheat your oven to 425°F (220°C) and line a small baking sheet with parchment paper.
Season the beef tenderloin on all sides with sea salt and black pepper.
Heat olive oil in a skillet over high heat and sear the beef for 1-2 minutes per side until browned; remove from pan and immediately brush with Dijon mustard.
In the same skillet, sauté finely minced mushrooms and shallots with fresh thyme until all liquid has evaporated and the mixture is dry.
Lay out a piece of plastic wrap, place the prosciutto slice on top, and spread the mushroom mixture evenly over the prosciutto.
Place the seared beef in the center and roll the prosciutto and mushrooms tightly around the beef using the plastic wrap; chill in the freezer for 5 minutes to set.
Roll out the puff pastry into a thin square, remove the plastic from the beef, and place the beef in the center of the pastry.
Fold the pastry over the beef, sealing the edges with a bit of water, and place seam-side down on the baking sheet.
Whisk the egg to create an egg wash, brush it over the pastry, and bake for 15-20 minutes until the pastry is deeply golden and the beef reaches your desired doneness.