Finely mince the cremini mushrooms, shallot, and garlic until they form a coarse paste.
Sauté the mushroom mixture in olive oil over medium heat until all moisture has evaporated and the mixture is dry, then stir in the fresh thyme.
Season the beef tenderloin with sea salt and black pepper, then sear in a hot pan for 1 minute on all sides until browned but raw in the center.
Brush the seared beef evenly with Dijon mustard while still warm and let it rest.
Lay a piece of plastic wrap on a flat surface, place the prosciutto slice down, spread the mushroom duxelles over it in an even layer, and place the beef in the center.
Roll the plastic wrap tightly to form a log and refrigerate for 15 minutes to set the shape.
Roll out the puff pastry into a thin rectangle, place the chilled beef log in the center after removing the plastic, wrap the pastry around it, and seal the edges with a bit of water.
Brush the exterior of the pastry with the egg white and bake at 400°F for 20-25 minutes until the pastry is golden and the beef reaches your desired internal temperature.