Butter 4 (6-ounce) creme brulee ramekins with the butter and lay 1 slice of ham in each ramekin so that it sits flat on the bottom (it's ok if it comes up the sides.) Crack 2 eggs into 2 separate small cups and pour simultaneously into 1 of the ramekins so that the eggs sit side by side on top of the ham slice. Season lightly with salt and pepper. Repeat with remaining eggs. Place the 4 filled ramekins onto a baking sheet and transfer to the oven. Bake until eggs are just beginning to set up and become opaque, 9 to 10 minutes.
Remove from the oven and drizzle each ramekin with 2 tablespoons of the heavy cream and then sprinkle 2 tablespoons of the cheese over the top of each. Sprinkle the parsley and chives over the ramekins and then return to the oven. Cook until eggs are just set up but yolks are still runny, 10 to 12 minutes, or to desired degree of doneness. Remove from the oven and serve immediately.