Melt butter in a large skillet over medium heat. Add the onion, garlic, rosemary, and paprika and cook until the onion has softened, 4-5 minutes. Stir in the cream cheese and chicken broth and continue to cook until the cream cheese melts. Stir in the chicken and ham and cook until just warmed through.
Remove pan from the heat and add the Swiss cheese, tossing to combine. Stir in the parsley. Season with the salt and pepper. Divide the mixture among 4 8-oz individual ramekins or soufflé dishes.
Unfold the thawed puff pastry on a lightly floured surface. Using a floured rolling pin, roll the pastry out to a thickness of 1/4''. Cut out 8 circles slightly larger in diameter than the ramekins being used. Whisk the egg and milk together in a small bowl. Dab the rim of the ramekins with the egg wash before placing one circle of pastry over the crock. Lightly pinch the edges to adhere. Brush the puff pastry with the egg wash. Bake until the pastry is a golden brown, 12-15 minutes. Let rest for 5 minutes before serving.
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