Generously season beef with salt and pepper. Heat 1 tablespoon oil in large skillet over high; sear all sides 1-2 minutes each until browned. Remove; cool completely.
Add remaining oil to same skillet; cook mushrooms, shallots, and garlic over medium-high 8-10 minutes until liquid evaporates. Add thyme and wine; cook until wine reduces. Season with salt and pepper; cool completely.
Heat oven to 400°F; line sheet with parchment.
Top prosciutto halves with spoonful mushroom mixture and beef piece; wrap prosciutto around.
Cut each pastry sheet into 6 squares. Center wrapped beef on each; fold pastry around, sealing with egg wash. Place seam-side down on sheet.
Brush with remaining egg; sprinkle flaky salt. Cut small steam vents in tops.
Bake 20-25 minutes until golden and puffed. Rest 5 minutes; serve warm.