Heat the oven to 375° F. Line three large baking sheets with parchment paper. Beat the egg with 1 tablespoon of water in a medium bowl. Mix the salt and paprika in a small bowl.
Unfold one sheet of puff pastry on a lightly floured work surface. If the frozen pastry cracks at the seams, use a rolling pin to roll over the seams to seal it shut. Brush the egg mixture onto the surface of the pastry, then top with ¼ cup of the parmesan, ½ cup of the manchego, half of the paprika salt, and 1½ teaspoons sesame seeds. Using the weight of the rolling pin, lightly roll the seasonings into the puff pastry. Using a 1½-inch star-shaped cookie cutter, slice the pastry into stars as closely as possible and place them on a prepared sheet 1 inch apart. Place the scraps on another sheet.
Bake stars until lightly browned and puffed, 10 to 15 minutes. Repeat with the second pastry sheet, the remaining ingredients and the second baking sheet. After the second sheet is finished, bake the scraps. Let the crisps cool completely on the baking sheets set on wire racks.
Storage tip: These puffs taste best the day they’re made.
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