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Gather all ingredients. Preheat the oven to 350 degrees F (180 degrees C). Line a baking sheet with parchment.
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Roll puff pastry out onto a lightly floured work surface. Using a rolling pin or your fingers, roll or pat pastry into a square. Cut pastry vertically into 3 rectangles. Transfer to the prepared parchment.
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Crack eggs into a bowl and whisk with basil, salt, and pepper. Add half the cheese and stir to combine.
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Add egg mixture to a nonstick skillet over medium heat. Stir eggs until just combined and cooked through. Remove from heat.
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Add a scoop of eggs on one side of each puff pastry rectangle. Top eggs with a sprinkle of cheese, then some bacon.
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Carefully lift the other side of pastry rectangle and fold it over the side with filling. Use the back of a fork, press down all around the edges to seal completely. Brush each pie with melted butter. Sprinkle with remaining cheese.
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Bake in the preheated oven until puffed and golden, about 15 minutes.