Preheat your oven to 400°F.
Heat olive oil and butter in a large pan over high heat.
Season the beef fillet generously with salt and black pepper.
Sear in the hot pan for about 4 minutes, turning to seal all sides, then remove and let rest.
In the same pan, lower the heat to medium and add the chopped mushrooms and rosemary.
Cook until mushrooms are soft and any liquid has evaporated.
Remove from heat and let them cool.
Place a sheet of plastic wrap on a flat surface and lay prosciutto slices in a single layer.
Spread the cooled mushroom mixture over the prosciutto.
Brush the cooled beef fillet with Dijon mustard and place on the prosciutto layer.
Wrap it tightly with the plastic wrap and refrigerate for 15 minutes.
Roll out the puff pastry, remove the beef from the plastic wrap, and lay it on the pastry.
Fold the pastry over the beef, sealing the edges with a little water.
Transfer the wrapped beef to a baking tray seam-side down.
Beat the egg and brush over the pastry for a nice glaze.
Bake for 20-25 minutes until the pastry is golden and flaky.