01 - Pat the beef tenderloin dry with paper towels. Season generously on all sides with salt and pepper. Heat vegetable oil in a large heavy skillet over high heat until smoking. Sear beef on all sides until deeply browned, approximately 2-3 minutes per side. Transfer to a plate and let cool completely to room temperature, about 20 minutes.
02 - In the same skillet, melt butter over medium heat. Add shallots and garlic; sauté until softened and fragrant, about 2 minutes. Add chopped mushrooms and thyme. Cook, stirring frequently, until mushrooms release their moisture and the mixture reduces to a thick, dry paste, 10-15 minutes. Season with salt and pepper. Transfer to a bowl and cool completely.
03 - Lay a large sheet of plastic wrap on a clean work surface. Arrange prosciutto slices in slightly overlapping rows to form a rectangle approximately 12x10 inches.
04 - Spread cooled mushroom duxelles evenly over the prosciutto layer. Place seared beef in the center. Using the plastic wrap as an aid, tightly roll the prosciutto and mushrooms around the beef to encase it completely. Twist the plastic wrap ends tightly to secure into a uniform log shape. Refrigerate for at least 30 minutes to firm.
05 - On a lightly floured surface, roll puff pastry into a rectangle approximately 14x10 inches. Unwrap the chilled beef log and place it in the center of the pastry.
06 - Whisk egg yolk with milk to create an egg wash. Brush the edges of the pastry with egg wash. Fold the long sides of pastry over the beef, overlapping to seal. Fold in the short ends and press firmly to seal. Trim any excess pastry. Place seam-side down on a parchment-lined baking sheet.
07 - Brush the entire surface with remaining egg wash. Use a sharp knife to score decorative patterns in the pastry if desired. Refrigerate for 15 minutes to firm the pastry.
08 - Position oven rack in the center position. Preheat oven to 425°F.
09 - Bake for 40-45 minutes until pastry is deep golden brown and crispy. An instant-read thermometer inserted into the center should read 120-125°F for medium-rare. Remove from oven and let rest for 15 minutes before carving—the internal temperature will rise to 130-135°F.
10 - Using a sharp serrated knife, slice the Wellington into 1-inch thick portions. Serve immediately with Dijon mustard on the side. Accompany with roasted potatoes, green beans, and a full-bodied red wine such as Bordeaux or Cabernet Sauvignon.