Heat oil in a large skillet on medium-high. Season beef with ¾ tsp each salt and pepper, then cook until browned on all sides, 5 to 7 minutes total. Transfer to a plate, let rest 5 minutes, then remove twine and brush with mustard.
Wipe out skillet, then add butter and heat on medium-low. Add shallots and cook, stirring occasionally, 4 minutes.
Meanwhile, to a food processor, add mushrooms and pulse to finely chop. Increase heat under skillet to medium, add mushrooms to shallots and cook, stirring occasionally, until mushrooms have released their liquid, about 5 minutes. Continue cooking, stirring often, until mushrooms have browned, 12 to 15 minutes more.
Stir in garlic and thyme and cook 2 minutes. Stir in sherry and cook until nearly dry, 15 seconds. Transfer to a plate.
Overlap 3 sheets of plastic wrap on a work surface. Place prosciutto on top, overlapping to make roughly a 12- by 7-inch rectangle. Spread mushroom mixture evenly over prosciutto.
Place beef tenderloin on the bottom edge of mushrooms and carefully roll tenderloin in mushroom-prosciutto layer, using the plastic wrap to help tighten it as you roll. Once it is completely rolled, wrap in plastic, twisting the ends to make sure it is very tight. Refrigerate 15 minutes.
Heat oven to 400°F. On a lightly floured surface, unfold puff pastry and roll to a rectangle about 3 inches wider than the beef roll on the shortest side.
Unwrap beef, and place log along the edge of the long side of pastry. Brush edges of puff pastry with beaten egg. Roll the beef in the puff pastry to cover completely. Fold ends of puff pastry like a present to cover ends and pinch to seal and folding underneath. Place on baking sheet seam side down.
Brush top and sides of pastry with egg, then use a small sharp knife to cut a few slits into the top of the pastry. Roast until an instant-read thermometer inserted in the center reads 120°F degrees for rare, 30 to 35 minutes. Let rest 10 minutes, then slice and serve while warm.