Season beef with salt and pepper. Sear in hot oil until browned on all sides. Let cool, then brush with Dijon mustard.
In a skillet, cook butter, shallots, garlic, and mushrooms until moisture evaporates, about 15 minutes. Let cool.
Lay out plastic wrap, then layer prosciutto and mushroom mixture. Place beef on top and roll into a tight log. Chill for 30 minutes.
Roll out puff pastry. Unwrap beef and place in center. Wrap and seal edges. Brush with egg wash.
Chill 15 minutes. Preheat oven to 400°F (200°C).
Bake 35–40 minutes or until golden and internal temperature reaches 125–130°F for medium-rare.