1. To make the Mushroom Duxelles: Heat the butter in a medium skillet over medium-high heat. Add the shallots and garlic and cook, stirring, for 30 seconds. Add the mushrooms, salt, and white pepper, reduce the heat to medium, and cook, stirring, until all the liquid has evaporated and the mushrooms begin to caramelize, about 12 minutes. Add the wine and cook, stirring to deglaze the pan, until all the liquid has evaporated. Remove from the heat and let cool before using.
2. Preheat the oven to 425 degrees F. Line a baking sheet with parchment paper and set aside.
3. Season both sides of each filet with ¼ teaspoon of the salt and 1/8 teaspoon of the pepper.
4. Heat the oil in a large heavy skillet over medium-high heat. Add the filets and sear for 1 minute on each side for medium-rare. Transfer to a plate to cool completely.
5. Roll out the puff pastry on a lightly floured surface to a 14-inch square, and cut into 4 (7-inch) squares.
6. Spread one-quarter of the mushroom duxelles on top of each filet and top each with 1 slice of the pate, pressing to flatten. Place one filet, mushroom side down, in the center of a puff pastry square. Using a pastry brush or your finger, paint the inside edges of the pastry with egg wash. Fold the pastry over the filet as though wrapping a package and press the edges to seal. Place the packages seam-side down on the prepared baking sheet. Brush the egg wash over the tops and sides of each package and bake until the pastry is golden brown and an instant-read thermometer registers 140 degrees F for medium-rare, about 20 minutes.
7. Remove from the oven and let rest for 10 minutes before serving.