Preheat the oven to 400°F. Line a baking sheet with parchment paper or a Silpat
Place the sheet of puff pastry on a lightly floured work surface and cut out four circles about 4 1/2 inches in diameter each. (See Kelly's Note.) Transfer the puff pastry circles to the baking sheet, spacing them about 2 inches apart, and score a circle border about 1/4 inch from the edge of each circle. Prick the centers then refrigerate the puff pastry for 10 minutes.
Remove the baking sheet from the fridge and transfer it immediately into the preheated oven. Bake the puff pastry for 8 minutes until the edges have risen.
Remove the puff pastry from the oven, sprinkle the cheese in the centers then crack an egg into each circle. Sprinkle the eggs with salt and pepper then bake for an additional 12 minutes until the egg whites are fully cooked. Remove the baked eggs, top with the chives and serve.