Season the roast with salt and pepper. In a large (preferably iron) skillet, heat 2 T of butter, over medium-high heat. When the butter is hot, (but not browning) brown the meat on all sides, being careful to just sear it. This should only take a minute or two for each “side”. You want a nicely browned crust on the meat.
Cover the meat and refrigerate for at least an hour, until the meat is cold.
Meanwhile, heat the remaining 2T of butter in the skillet, over medium-low heat. Once melted, add the minced onion and sauté until softened, stirring constantly. Add the mushrooms and sauté until softened, stirring. Add the vermouth to the onion and mushroom mixture and reduce until almost absorbed. Add the liver pate and turn off the heat. Mash the pate and the mushroom mix together. Allow to cool.
Sprinkle one sheet of the pastry with flour and roll out large enough to enclose the roast tenderloin completely. (The other sheet can be re-frozen or used to make decorative shapes for the top of your Wellington). Spread the liver/mushroom mix over the entire fillet. Place the cooled meat in the center of the pastry. Wrap the pastry around the meat and seal with a little of the egg wash.
Lift the roast onto a shallow baking dish or a cookie sheet, seam side down, and refrigerate at this point (Can be done the day ahead).
Pre-heat oven to 400°F Bake for 30 minutes or until the puff pastry is golden brown. Check the internal temperature of the beef. Reduce the heat to 350F and bake for 10 minutes or until the internal temperature of the meat is 120°F (If the pastry is browning too quickly, cover loosely with foil). Remove the roast from the oven, and let stand, uncovered, for 15 minutes. Carve using a serrated knife, into thick slices.
(This dish is best rare, and really, should be eaten this way. For well-cooked meat, I’d suggest cutting the roast into 4 thick steaks and following the above steps, only wrapping individually.).