Heat butter in a large skillet over medium heat. When butter is melted, add eggplant slices.
Sprinkle with salt and pepper and cook until tender and mostly translucent, about 3 to 5 minutes per side.
To poach the eggs, fill a large pot about half full with water. Bring to just a boil, then reduce heat so it is just simmering (I found that about 180F to 185F on an instant-read thermometer was perfect).
Break each egg individually into a fine-mesh sieve set over a bowl and let the watery part of the white drain out. Add egg to water and repeat with remaining eggs, cooking each about 3 minutes and gently turning over each egg once during cooking. Cook until white is set but the yolk remains soft.
Remove with a slotted spoon to let water drain off.
To assemble, top each eggplant slice with ham or bacon.
Top this with a poached egg, then drizzle with hollandaise (if the sauce has thickened while cooking everything else, simply add a tsp or so of warm water and whisk briskly).
Season with salt and pepper to taste and sprinkle with paprika.