Prep the Beef Tenderloin: Wrap the beef tightly in a triple layer of plastic wrap, shaping it into a neat cylinder. Refrigerate overnight if possible, or at least 30 minutes.
Make the Mushroom Duxelles: Heat the olive oil in a pan over medium heat. Add the finely diced mushrooms, then season with salt, pepper, and thyme. Cook until all the moisture has released and fully evaporated. The pan should look dry. Set aside and let cool completely.
Make the Chive Crepes: In a large bowl, whisk together the egg, flour, milk, chives, thyme, and salt until smooth. Refrigerate the batter for at least 30 minutes. Heat a nonstick pan over low heat and lightly coat with cooking spray. Add ¼ cup batter, swirling it into a thin layer. Cook for 25 to 30 seconds until just set, then flip and cook for a few more seconds. Keep the crepes pale, not browned. Repeat as needed.
Sear and Season the Beef: Remove the plastic wrap and pat the beef dry. Season all sides with salt and pepper. Heat a cast iron skillet over high heat until smoking hot. Add the grapeseed oil, then sear the beef on all sides, including the ends, for 60 seconds per side. Transfer to a tray, pour any juices from the pan over it, and brush all over with Dijon mustard while still hot.
Assemble the First Wrap: Moisten your work surface and lay out three overlapping pieces of plastic wrap. Trim 2 crepes into squares that match the length of the beef and lay them on the plastic wrap. Arrange the prosciutto over the crepes, then spread the cooled duxelles evenly over the prosciutto. Place the beef in the center. Using the plastic wrap, roll everything tightly into a cylinder. Twist the ends to secure and refrigerate for 20 minutes.
Wrap in Puff Pastry: Lay out three more overlapping pieces of plastic wrap. Roll the puff pastry into an ⅛-inch-thick rectangle. Place the chilled beef roll in the center lengthwise. Brush the pastry edges with egg wash, then roll the pastry tightly around the Wellington, keeping the seam on the bottom. Refrigerate for 30 minutes.
Bake the Wellington: Preheat the oven to 425°F convection or 450°F regular bake. Unwrap the Wellington, brush all over with egg wash, and place seam side down on a parchment-lined baking tray. Bake until the pastry is deeply golden and the center reaches 135°F for medium-rare, about 20 to 40 minutes depending on your oven and the starting temperature of the beef.
Rest and Slice: Let the Wellington rest for 15 to 20 minutes before slicing. Use a sharp serrated knife to cut into thick slices, about 1 inch each.
Heat a cast iron skillet over high heat until it’s smoking hot. Add a little oil with a high smoke point (such as avocado or grapeseed oil). Place the beef in the pan and sear each side for 60 seconds, including the ends.
Once seared, transfer the beef to a tray and pour any juices from the pan over it. While the beef is still hot, brush it all over with Dijon mustard. Set aside to rest.