Pat the filet mignon steaks dry with paper towels and season all over with salt and pepper.
Heat olive oil in a large skillet over high heat. Once the oil is shimmering, sear the steaks for 2 minutes per side until a brown crust forms. Transfer the steaks to a plate and let them cool completely.
Reduce the heat to medium. Add the butter to the same skillet. Once melted, add the chopped mushrooms, shallots, and thyme. Cook, stirring occasionally, for 8-10 minutes, until the mushrooms have released their liquid and the mixture is dry. Remove from the heat and let it cool completely.
On a lightly floured surface, roll out the puff pastry sheet into a 14x14-inch square. Cut the pastry into four equal 7x7-inch squares.
Spread 1/4 of the garlic and herb cheese over the top of each cooled steak. Top the cheese with 1/4 of the cooled mushroom mixture.
Place one steak in the center of each pastry square. Lightly brush the edges of the pastry with the beaten egg wash. Fold two opposite corners up over the steak, then bring the other two corners up to meet in the middle. Pinch all the seams together firmly to seal the package.
Place the wrapped wellingtons seam-side down on a parchment-lined baking sheet. Cover and refrigerate for at least 1 hour or up to 24 hours.
Preheat your oven to 425°F (220°C). Remove the wellingtons from the refrigerator, unwrap, and brush the entire surface of each with the egg wash.
Bake for 15-18 minutes, or until the pastry is golden brown and puffed. For medium-rare, the internal temperature should be 130-135°F (54-57°C). Let the wellingtons rest for 5-10 minutes before serving.