Season the beef with sea salt and black pepper, then sear in olive oil over high heat for 1 minute per side until browned but raw inside.
Remove the beef from the pan and immediately brush with Dijon mustard while warm, then set aside to cool completely.
Finely mince the mushrooms, shallot, and garlic, then sauté in the same pan with fresh thyme until all moisture has evaporated and the mixture is dry.
Lay out a piece of plastic wrap, place the prosciutto slices slightly overlapping, and spread the mushroom duxelles evenly over the prosciutto.
Place the beef in the center of the mushrooms, roll tightly using the plastic wrap to form a uniform log, and chill in the refrigerator for 15 minutes.
Roll out the puff pastry on a lightly floured surface, unwrap the beef log and place it on the pastry, then fold the pastry over and seal the edges tightly.
Brush the exterior with a beaten egg wash and bake at 425°F for 20 minutes until the pastry is golden and the internal temperature of the beef reaches 130°F.