Preheat oven to 350 degrees. Line 2 baking sheets with parchment paper; set aside.
Place one sheet of puff pastry on a lightly flowered surface. Sprinkle surface with 2 tablespoons of the cheese. Press cheese into dough with your fingers. Cover the sheet of pastry with the remaining sheet, sandwiching the cheese in between. Roll the pastry into an 18-by-20 inch rectangle. Sprinkle the pastry with half the remaining cheese and the basil. Press cheese and the herbs into the pastry with your fingers. Fold pastry in half crosswise. Re-roll the pastry into an 18-by-20-inch rectangle.
Beat egg yolk and water together with a fork until combined. Lightly brush pastry with egg glaze and sprinkle the remaining cheese. Cut pastry into 40 strips, ½-by-18 inches. Twist each strip several times on the baking sheet.
Bake cheese straws in the preheated oven until golden brown, 12-15 minutes.