Finely mince the mushrooms, shallot, and garlic. Sauté them in a dry pan over medium heat with thyme until all moisture has evaporated and the mixture is paste-like, then set aside to cool.
Season the beef tenderloin with sea salt and black pepper.
Heat olive oil in a skillet over high heat and sear the beef for about 1 minute on all sides until browned but raw in the center. Remove and let cool.
Lay a piece of plastic wrap on a flat surface. 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 tightly roll the prosciutto and mushrooms around the meat. Chill in the refrigerator for 15 minutes to set the shape.
Preheat your oven to 400°F (200°C).
Roll out the puff pastry on a lightly floured surface. Remove the beef from the plastic wrap and place it on the pastry.
Wrap the pastry around the beef, trimming any excess, and seal the edges with a bit of beaten egg.
Place the Wellington on a parchment-lined baking sheet, seam-side down. Brush the entire exterior with the remaining beaten egg wash.
Bake for 20-25 minutes until the pastry is deep 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.