Preheat oven to 425°F.
Place the beef tenderloin on a baking sheet lined with parchment paper and rub with the olive oil, salt, and pepper. Roast for 30 minutes or until an instant-read thermometer reads 120°F when inserted into the center of the meat. Transfer to a cutting board to cool.
Heat the butter and mushrooms in a large skillet over medium heat until the mushrooms are softened, about 6 minutes. Add the shallots and thyme and cook for 2 more minutes. Add the sherry, stir to combine, then remove from heat and set aside to cool.
Place 12 slices of prosciutto on a cutting board or baking sheet lined with parchment paper in two strips of six slices, each overlapping slightly at one end. Spread the cooled mushroom mixture over top of the prosciutto slices, leaving a 1-inch border along all sides of the prosciutto slices.
Place beef tenderloin on top of the mushroom mixture and fold prosciutto slices up and over beef to completely cover. Place entire beef tenderloin in the center of puff pastry dough. Lightly brush edges of puff pastry with egg wash, then fold up sides to completely wrap beef tenderloin. Trim any excess dough and discard.
Brush top and sides of puff pastry with remaining egg wash, sprinkle with flaky sea salt and black pepper, and cut two small slits for venting on the top of the pastry. Bake in preheated oven for 30 minutes or until internal temperature is 135°F when inserted into the thickest part of the meat. Let rest 10 minutes before slicing into ½-inch slices and serving warm.