Season beef pieces generously with salt and pepper. Heat 1 tablespoon olive oil in a large skillet over high heat and sear beef on all sides until browned, about 1-2 minutes per side. Remove and let cool completely.
In the same skillet, add remaining olive oil and cook mushrooms, shallots, and garlic over medium-high heat until all liquid has evaporated, about 8-10 minutes. Add thyme and red wine, cooking until wine reduces. Season with salt and pepper, then let cool completely.
Preheat oven to 400°F and line a baking sheet with parchment paper.
Lay out prosciutto halves and top each with a spoonful of mushroom mixture. Place a beef piece on top and wrap the prosciutto around it.
Cut each puff pastry sheet into 6 squares. Place a prosciutto-wrapped beef in the center of each square and fold the pastry around it, sealing edges with egg wash. Place seam-side down on prepared baking sheet.
Brush tops with remaining egg wash and sprinkle with flaky sea salt. Cut small slits in the top for steam to escape.
Bake for 20-25 minutes until pastry is golden brown and puffed. Let rest for 5 minutes before serving warm.