Butter the Pie Pan with the butter and line the pan with the ham slices, so that the bottom and sides of the Pie Pan are completely covered. Place the pan into the basket of the Air Fryer. 

Crack 1 of the eggs in a small bowl and add the heavy cream, ¼ teaspoon of the salt, and ⅛ teaspoon of the black pepper. Whisk well to combine.  Pour egg mixture into the Pie Pan, over the ham, and crack the remaining 3 eggs over top.  Season with the remaining salt and pepper and sprinkle with the Parmesan cheese. Place basket into the drawer of the Air Fryer, and insert into the machine. Set the temperature to 320 degrees Fahrenheit and the timer to 12 minutes. 

When the timer goes off, remove the Pie Pan from the Air Fryer. Season with the Pimenton and garnish with the shaved chives. Use a spatula to remove the shirred eggs from the Pie Pan, and transfer to a plate. Serve warm.