Preheat the oven to 375 degrees.
Roll out each sheet of puff pastry on a lightly floured board until it’s 10 3 12 inches. Beat the egg with 1 tablespoon of water and brush the surface of the pastry. Sprinkle each sheet evenly with 1/4 cup of the Parmesan, 1/2 cup of the Gruyère, 1/2 teaspoon of the thyme, 1/2 teaspoon of the salt, and some pepper. With the rolling pin, lightly press the flavorings into the puff pastry. Cut each sheet crosswise with a floured knife or pizza wheel into 11 or 12 strips. Twist each strip and lay on baking sheets lined with parchment paper.
Bake for 10 to 15 minutes, until lightly browned and puffed. Turn each straw and bake for another 2 minutes. Don’t overbake or the cheese will burn. Cool and serve at room temperature.
You want to work quickly because the puff pastry needs to stay cold until it’s baked. You can also brush the pastry with pesto, tapenade, or sun-dried tomato paste instead of sprinkling with the cheeses.
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