Put 2 cups diced potatoes in a medium pot and cover with cold water. Bring to a boil over high heat, then boil about 10 minutes or until the potatoes are just fork-tender.
Drain the potatoes thoroughly and set them aside to drain/dry for a few minutes.
Heat a large cast-iron (or heavy) skillet over medium heat. Add 1 tablespoon olive oil and 2 tablespoons butter and allow the butter to melt.
Add 1 medium onion, finely diced, to the skillet. Cook, stirring occasionally, until the onions soften and begin to caramelize, about 10 minutes.
Add the drained potatoes to the skillet. Spread them into an even layer and cook, stirring and flipping occasionally, until the potatoes begin to brown, about 8–10 minutes. Season with salt and black pepper to taste.
Stir in 2 cups thinly sliced cooked prime rib and cook just until the meat is warmed through, about 1–3 minutes. Remove the skillet from the heat.
While the hash finishes, cook 4 large eggs sunny-side up, fried, or poached according to your preference.
Divide the hash among plates, top each portion with one cooked egg, garnish with fresh chopped parsley, and serve immediately.