Finely mince the mushrooms, shallot, and garlic in a food processor, then sauté in a dry pan with thyme until all moisture has evaporated.
Season the beef tenderloin with sea salt and black pepper, then sear in avocado oil over high heat for 60 seconds per side until browned.
Immediately brush the warm beef with Dijon mustard on all sides and set aside to rest.
Lay a sheet of plastic wrap on the counter, place the prosciutto slice down, spread the mushroom duxelles over it, and place the beef in the center.
Roll the beef tightly in the prosciutto and plastic wrap to form a log, then chill in the refrigerator for 15 minutes to set the shape.
Roll out the puff pastry thinly, wrap it around the chilled beef log, seal the edges, brush with beaten egg wash, and bake at 425°F for 20 minutes until golden.