Preheat your oven to 400°F (200°C).
Season the beef fillets with salt and pepper. Heat olive oil in a skillet over high heat and sear the fillets for 1-2 minutes on each side. Set aside to cool.
In the same skillet, add the mushrooms and cook until they release their moisture and become dry. Set aside to cool.
Lay out the prosciutto slices on a sheet of plastic wrap. Spread the mushroom mixture evenly over the prosciutto. Place a beef fillet on top and roll up tightly using the plastic wrap. Chill in the fridge for 15 minutes.
Roll out the puff pastry on a floured surface. Unwrap the beef fillets and place them in the center of the pastry. Brush the edges with beaten egg and fold the pastry over the beef, sealing the edges.
Place the wrapped fillets on a baking sheet, seam side down. Brush the tops with more beaten egg.
Bake in the preheated oven for 25-30 minutes, or until the pastry is golden brown and the beef is cooked to your liking.
Let rest for 10 minutes before slicing and serving.