Heat olive oil in a heavy skillet over high heat until very hot. Add filets; sear until well browned, 1 to 2 minutes per side. Transfer filets to a plate, cool slightly, then chill in the refrigerator for at least 1 hour. Filets must be cold.

Melt butter in the same skillet over medium heat. Add mushrooms, shallots, and garlic; cook and stir until shallots are tender and translucent and mushrooms release their juices, 6 to 8 minutes. Stir in 2 tablespoons wine, scraping any browned bits of food off the bottom of the skillet. Transfer mushroom mixture to a bowl, cool slightly, then chill in the refrigerator, about 45 minutes.

Place 2 puff pastry squares on a work surface; top each with 1 chilled filet. Divide mushroom mixture between filet tops; cover each with 1 remaining puff pastry square. Fold and pinch pastry edges together to enclose filling and seal. Trim pastry to make tidy packages; cut a small slit on top each package.

Heat remaining 1 1/2 cups red wine in a saucepan over medium heat; simmer until wine is reduced by half, about 15 minutes. Season with salt and black pepper; set aside and keep warm.

Whisk egg and milk together; brush pastry packages with egg mixture to help browning. Refrigerate pastry packages to keep cold.

Preheat the oven to 450 degrees F (230 degrees C). Line a baking sheet with parchment paper.

Place puff pastry packages on the prepared baking sheet. Bake in the preheated oven until pastry is golden brown and crisp and filets reach desired doneness, about 15 minutes for medium-rare. An instant-read thermometer inserted into centers should read 130 degrees F (54 degrees C). Drizzle wine sauce over beef in puff pastry to serve.