How to Make It
In a food processor, pulse the ham until finely chopped. Transfer the ham to a bowl and stir in the mayonnaise, mustard, hot sauce, cayenne and chives.
Toast the bread until golden. Spread the toast with 2 teaspoons of the butter and top with the deviled ham.
In a very large nonstick skillet, melt the remaining 1 tablespoon of butter in the olive oil. Crack the eggs into the skillet and cook over moderately high heat until the whites are set, about 2 minutes. Season each egg lightly with salt and black pepper. Using a spatula, set the eggs on the deviled ham sandwiches and serve immediately.