Heat 1 teaspoons oil in a medium skillet over medium-high heat. Season beef on all sides with half the salt and pepper. Sear in a skillet, cooking 2 minutes on each side until browned. Set aside on a towel-lined plate and refrigerate to cool.
Using a knife or food processor, finely chop mushrooms. In the same skillet, heat remaining oil and butter over medium-high heat. Add shallot and sauté until fragrant. Add mushrooms, remaining salt and pepper and sauté, stirring constantly, until mushrooms give up their liquid and brown slightly. Stir in thyme leaves, transfer to a small bowl, and cool completely.
Place prosciutto slice on a 12-inch piece of plasric wrap and spread evenly with mushroom mixture. Dry bottom of meat with paper towel and place it in the center. Use the plastic wrap to roll into a cylinder. Twists ends together and refrigerate.
Dust worktop with flour and roll puff pastry into a rectangle about 1/8-inch thick. Unwrap beef and place in the center of the pastry. Beat the egg and water together. Paint the surrounding dough with some of the egg wash and wrap around beef, trimming extra dough away and tucking in ends,.
Place seam side down on parchment paper-lined baking tray, cover loosely and chill at least 20 minutes, or up to 2 hours.
Preheat oven to 425°F Paint chilled pastry wirh remaining egg wash. Cut three diagonal slits across the top and sprinkle with kosher salt. Roast on the second rack from the bottom for 45 minutes for medium, or until an instant-read thermometer reaches 160°F Let the meat rest for at least 10 minutes before cutting into a 1-inch slices to serve.