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

Melt butter in the same skillet over medium heat. Add mushrooms, shallots, and garlic. Cook and stir until shallots are tender and mushrooms release juices, 6-8 minutes. Stir in 2 tablespoons red wine and scrape browned bits from the skillet. Transfer mixture to a bowl and chill in the refrigerator for about 45 minutes.

Place two puff pastry squares on a work surface, top each with a chilled filet, and distribute the mushroom mixture on top. Cover each with an additional puff pastry square, fold and pinch edges to seal. Trim excess pastry and cut a small slit in each package.

Heat remaining 1 ½ cups red wine in a saucepan over medium heat, simmer until reduced by half (about 15 minutes). Season with salt and pepper and keep warm.

Whisk egg and milk together and brush the pastry packages with the mixture for browning. Refrigerate to keep cold before baking.

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

Place the pastry packages on the baking sheet and bake in the preheated oven for approximately 15 minutes, or until golden brown and reaching medium-rare doneness. An instant-read thermometer should register 130°F (54°C).