Heat 1 tsp (5 mL) 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 min. on each side until browned. Set aside on a towel-lined plate and refrigerate to cool.
Using a knife or a food processor, finely chop mushrooms. In the skillet used for searing, heat remaining oil and butter over medium-high heat. Add shallots and sauté until fragrant. Add mushrooms and remaining salt and pepper. Cook, stirring, until mushrooms give up their liquid and brown slightly. Stir in thyme leaves, transfer to a bowl to cool completely.
Place prosciutto slices on a large piece of plastic wrap, slightly overlapping if needed, to align to the same length as the beef tenderloin. Spread the mushroom mixture evenly on the prosciutto. Pat the beef dry. Place it on top of the mushrooms and prosciutto. Wrap tightly in the plastic wrap, twisting ends closed. Chill.
Dust worktop with flour and roll puff pastry into a rectangle about 1/8-in. (3 mm) thick. Unwrap beef roll and place in the centre of the pastry. Fork-beat the egg with 1 tbsp (15 mL) water. Paint the edges of the dough around the beef with egg wash, reserving leftover wash. Wrap the pastry around the beef roll. Trim off excess dough, tuck the ends of the pastry under. Place seam-side down on a parchment paper-lined baking tray. Cover loosely; chill at least 20 min. or up to 2 hrs.
Preheat oven to 220°C (425°F). Brush remaining eggwash on the chilled Beef Wellington. Cut three diagonal slits across the top for venting. Sprinkle pastry with kosher salt. Roast on the second rack from the bottom for 45 min. for medium doneness, or until an instant-read thermometer reaches 71°C (160°F) when inserted into the centre. Rest 10 min. before cutting into 1-in. (2.5 cm) slices to serve.