1. Tie the tenderloin in 4 places using kitchen twine so it is in a cylinder shape. Season all over with salt and pepper.
2. Coat the bottom of a heavy skillet with olive oil. Heat over medium-high heat until oil is shimmering. Sear tenderloin until well-browned on all sides, including the ends. Place the beef on a plate and cut off the twine. Let cool slightly and coat beef with dijon. Place in refrigerator to cool while preparing the mushrooms.
3. In a food processor, pulse the mushrooms, shallots, garlic and thyme until finely chopped. Melt 2 t. Butter in a skillet over medium heat. Add mushroom mixture and cook until all of the liquid has evaporated (10 – 20 minutes). Place in refrigerator to cool.
4. Place 2 sheets of plastic wrap (about 18” in length) on a work surface, overlapping them lengthwise down the center by about 6 inches. Shingle the prosciutto on the plastic wrap in 2 rows of 6 pieces each (large enough to cover the entire tenderloin). Spread the mushroom mixture evenly and thinly over the prosciutto. Season with salt and pepper.
5. Place the tenderloin at one end of the prosciutto. Roll up tightly in the plastic wrap, tucking in the ends of the prosciutto as you roll, so the whole beef is encompassed. Twist the ends of the plastic to seal the meat into a nice log shape. Refrigerate for 30 – 45 minutes.
6. Pre-heat the oven to 425F. Flour a work surface and roll out the puff pastry into a rectangle slightly bigger than the rectangle you made with the prosciutto and about ¼” thick. Depending on the brand of puff pastry, you might have to overlap 2 sheets and press them together.)
7. Set the beef in the center of the pastry. Fold over the long sides, brushing with beaten egg to seal. Trim the ends as necessary, then brush them with beaten egg and fold over to completely seal the beef. Crimp the edges with a fork to seal well. Wrap entire log in plastic wrap to form a tight cylinder and refrigerate for 20 minutes.
8. Line a baking sheet with parchment or foil. Place beef, seam side down on the pan. Brush with beaten egg and sprinkle with flaky or coarse sea salt. Cut a couple of slits in the top with a paring knife to allow steam to escape while baking.
9. Bake for 40-45 minutes until pastry is golden brown and the center of beef reaches 120-125f on an instant read thermometer. Remove from oven and let rest 10-15 minutes prior to slicing.