Finely mince the mushrooms, shallot, and garlic. Sauté them in a pan with thyme and a pinch of salt until all moisture has evaporated and the mixture is a thick paste.
Season the beef tenderloin with sea salt and black pepper. In a hot pan with olive oil, sear the beef on all sides for about 60 seconds per side until browned but raw in the center.
Brush the warm beef with Dijon mustard and set aside to cool completely.
Lay a piece of plastic wrap on the counter. Arrange the prosciutto slices in an overlapping rectangle, then spread the mushroom duxelles evenly over the prosciutto.
Place the beef in the center and use the plastic wrap to roll the prosciutto and mushrooms tightly around the meat. Chill in the fridge for 15 minutes to set the shape.
Roll out the puff pastry on a lightly floured surface. Remove the beef from the plastic and place it on the pastry. Wrap the pastry around the beef, sealing the edges with a bit of beaten egg.
Place the Wellington on a baking sheet, brush the entire exterior with the remaining beaten egg, and score the top lightly with a knife for decoration.
Bake at 425°F for 15-20 minutes until the pastry is golden brown and the internal temperature of the beef reaches 125°F for medium-rare.
Let the Wellington rest for at least 10 minutes before slicing into thick rounds to serve.