Preheat your oven to 425°F.
Season the beef tenderloin on all sides with the sea salt and black pepper.
Heat the olive oil in a heavy skillet over high heat and sear the beef for 2 minutes per side until a brown crust forms, then remove from heat.
While the beef is still warm, brush all sides with the Dijon mustard and let it rest.
Finely mince the cremini mushrooms and sauté them in the same skillet until all moisture has evaporated and they form a thick paste.
Lay a piece of plastic wrap on a clean surface and arrange the prosciutto in a slight overlap.
Spread the mushroom paste evenly over the prosciutto and place the beef in the center.
Roll the beef tightly in the prosciutto using the plastic wrap to form a log and chill in the freezer for 10 minutes to set the shape.
Roll out the puff pastry thinly, wrap it around the chilled beef log, and trim away any excess dough.
Place the wellington on a parchment-lined tray, brush the pastry with a beaten egg, and bake for 15-20 minutes until the crust is golden and crisp.