01 - Season the beef generously with salt and pepper. Heat the vegetable oil in a large skillet over high heat. Sear the beef on all sides until browned, about 2-3 minutes per side. Remove from heat and let cool completely.
02 - In the same pan, melt the butter over medium heat. Add the shallots and garlic; sauté until softened, about 2 minutes. Add mushrooms and thyme. Cook, stirring often, until mushrooms release moisture and mixture becomes dry and pasty, about 10-15 minutes. Season with salt and pepper. Set aside to cool.
03 - Lay a large sheet of plastic wrap on a work surface. Arrange the prosciutto slices in a slightly overlapping layer to form a rectangle large enough to wrap the beef.
04 - Spread the cooled mushroom duxelles evenly over the prosciutto. Place the cooled beef on top. Using the plastic wrap, roll the prosciutto and mushroom mixture tightly around the beef to form a log. Twist the ends of the wrap to secure. Chill in the refrigerator for at least 30 minutes.
05 - Roll out the puff pastry on a floured surface to a rectangle large enough to completely enclose the beef. Unwrap the beef log and place it in the center of the pastry.
06 - Mix the egg yolk with milk to make an egg wash. Brush the edges of the pastry with the egg wash. Fold the pastry over the beef, sealing the edges tightly. Trim excess pastry if needed. Place seam-side down on a baking sheet lined with parchment.
07 - Brush the top with egg wash. Use remaining pastry to decorate, if desired. Chill in the refrigerator for 15 minutes.
08 - Preheat oven to 425°F.
09 - Bake for 40-45 minutes, or until pastry is golden and a meat thermometer inserted into the center reads 122°F for medium-rare. Rest for 15 minutes before slicing.
10 - Serve with Dijon mustard, if desired.