Heat the oil in a large frypan over medium-high heat. Season the steaks with salt and pepper, then fry on each side until browned to lock in the juices. Remove from the frypan and set aside to cool slightly.
Add the mushrooms, wine and beef stock to the frypan; cook and stir over medium heat until mushrooms are tender. Use a slotted spoon to transfer the mushrooms to a small bowl, reserving the liquid for later. Mix the pate with the mushrooms, adding a little stock liquid if needed to achieve a spreadable texture. Divide the mushrooms among the steaks and spread on the top. Refrigerate for about 30 minutes.
Preheat the oven to 200 degrees C. Lay the puff pastry out on a clean surface and cut out two 15cm (approx) circles for each steak. You can roll it out a little thinner if you need more area. Place a steak onto a circle of pastry, and bring the edges up around the sides. Cover the top with a second piece of pastry, overlapping the edges and pressing to seal into a package. Repeat with remaining steaks. Place the Wellingtons onto a baking tray and brush the tops with beaten egg.
Bake for about 12 minutes, for rare, 16 minutes for medium, or 20 minutes for well done. Remove from the oven and allow to rest for about 5 minutes.
While the steaks are cooking, make the sauce. Melt the butter in a saucepan over medium heat. Stir in the flour and cook, stirring constantly, until browned. Gradually stir in some of the stock, and continue to cook and stir until the sauce thickens. Reduce heat to low and stir in the rest of the stock and then the wine. Continue to simmer until thickened, about 15 minutes. Remove from the heat and stir in butter and parsley. Season with salt and pepper to taste.
To serve, cut each Wellington in half and spoon some sauce onto the plate.