Season the steaks liberally with salt and pepper. Heat the olive oil in a skillet over high heat. Sear the steaks for 1 to 2 minutes per side until deeply browned. Remove from heat and brush immediately with Dijon mustard on all sides. Let the steaks cool completely.
Prepare the duxelles: Pulse the mushrooms, shallot, and thyme in a food processor until finely chopped but not pureed. Cook the mixture in the same skillet over medium heat until all moisture has evaporated, about 8 to 10 minutes. Stir in the sherry and cook until evaporated. Spread the duxelles on a plate to cool completely.
Lay out two large pieces of plastic wrap. Place two slices of prosciutto on each piece of plastic wrap, slightly overlapping. Spread half of the cooled duxelles evenly over the prosciutto slices.
Place one cooled steak at the edge of the prosciutto/duxelles layer. Use the plastic wrap to tightly roll the prosciutto and duxelles around the steak, forming a compact cylinder. Twist the ends of the plastic wrap to secure the shape. Refrigerate for at least 30 minutes (or up to 24 hours) to firm up.
Lightly flour a clean surface. Roll out the puff pastry sheet slightly thinner. Cut the pastry into two equal rectangles, large enough to enclose each steak.
Remove the plastic wrap from the chilled steaks. Brush the edges of the pastry rectangles with the beaten egg wash. Place one steak near the edge of a pastry rectangle. Roll the pastry around the steak, trimming excess pastry and sealing the edges firmly with a fork or your fingers to create a tight seal. Place seam-side down on a parchment-lined baking sheet.
Brush the entire surface of each Wellington generously with egg wash. Score the top lightly with the back of a knife in a decorative pattern, avoiding cutting through the pastry. Chill for 15 minutes.
Preheat your oven to 400 degrees Fahrenheit (200 degrees Celsius).
Bake for 20 to 25 minutes for medium-rare (internal temperature of 130-135°F or 54-57°C). Adjust time based on your desired doneness.
Let the Wellingtons rest on the baking sheet for 5 to 10 minutes before serving.
Notes
For best results, ensure the duxelles mixture is completely dry before wrapping the beef.
Use high-quality, all-butter puff pastry for the best flavor and crispness.
If you plan to make these ahead, assemble them completely, but do not egg wash. Cover and refrigerate for up to 12 hours, then egg wash and bake, adding 5 minutes to the baking time.