Make the Dutch baby: Preheat the oven to 425 degrees F. Melt the butter in a 12-inch cast-iron skillet or other large oven-safe skillet over medium heat.
In a large bowl, whisk the eggs and milk together until well combined. Add the flour, paprika, and salt and whisk until smooth. Pour the batter into the hot skillet and bake until the pancake puffs up around the edge and turns very golden brown, 20 to 25 minutes.
While the Dutch baby is baking, prepare the toppings: Melt the butter in a medium skillet over medium heat. Add the eggs and fry until the whites are set and the yolks are just set, 5 to 7 minutes. Season with salt and pepper.
When the Dutch baby has finished baking, remove it from the oven--it will begin to deflate shortly after it comes out. Garnish immediately, arranging the fried eggs and ham on top, then sprinkling with chives and a few pinches of smoked paprika.