How to Make It
Preheat oven to 375°F. Line 2 baking sheets with parchment paper.
Unfold 1 sheet of puff pastry on a lightly floured work surface, and roll out to a 10- x 16-inch rectangle. Brush pastry surface edge to edge with a small amount of beaten egg. Sprinkle evenly with 3/4 cup of the shredded Manchego cheese. Lay half of the prosciutto slices across lower half of pastry (along 10-inch side), overlapping slices slightly. Fold top portion of pastry over prosciutto with ends of the pastry meeting to form an 8- x 10-inch rectangle. Repeat with second puff pastry sheet.
Cut each rectangle into 20 (1/2- x 8-inch) strips. Working with 1 strip at a time, use both hands to twist and form a spiral stick. Place on prepared baking sheets, pressing ends onto parchment to adhere. Brush with remaining egg wash; sprinkle with thyme.
Bake in preheated oven until cheese straws are puffed and golden brown, 15 to 17 minutes.