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Preheat the oven to 400˚F. Line two sheet pans with parchment paper.
Gently roll each sheet of puff pastry to roughly 10 x 11 inches, then cut each sheet into 4 rectangles. Transfer the rectangles to prepared sheet pans.
Use a paring knife, score a ½ inch border around the edges of each rectangle. Don't cut all the way through.
Spread about 1 teaspoon of dijon mustard in an even layer in the center of each rectangle, then sprinkle with a small amount of fresh thyme. Top with a small handful of shredded cheese, then arrange the ham slices around the center, leaving a well for the egg.
Brush the exposed edges of each puff pastry rectangle with egg wash.
Bake for 10 minutes, until the cheese is melty and the puff pastry just starts to lightly brown. Remove from the oven. Use the back of a spoon to create a well in the center of each rectangle.
Crack an egg in the center of each pastry. Season with salt and pepper to taste.
Return to the oven for about 10 minutes, until the whites of the egg are cooked through and the yolk is still runny.
Garnish with chives, and enjoy warm!