How to Make It
Preheat the oven to 275°. Set a rack in a large baking sheet. In a large cast-iron skillet, heat 3/4 inch of olive oil to 300°. Working in 2 batches, fry the potatoes until golden brown and lightly crisp on the outside, about 7 minutes. Transfer the potatoes to the rack to drain and immediately season with kosher salt. Keep warm in the oven.
Carefully transfer 1 cup of the hot olive oil from the cast-iron skillet to a large nonstick skillet. Crack 4 of the eggs into the skillet and gently fry over moderately low heat until the whites are set and the yolks are slightly runny, about 5 minutes. Transfer to a plate. Repeat with the remaining 4 eggs.
Divide the fries among 4 plates. Top each plate with 2 fried eggs, 2 slices of ham and a spoonful of caviar, if desired, and serve immediately.