Place the potatoes in a small pot and cover with 1 inch cold water. Add a pinch of kosher salt and bring to a simmer; cook until the potatoes are tender, then drain and rinse under cold water to cool.
Preheat the oven to 450 degrees F.
While the potatoes are cooking, melt the butter in a saute pan and add the onions and celery. Season with salt and pepper and cook slowly until the vegetables soften. Stir in the parsley and rosemary, and then cool.
Combine the cooled potatoes, vegetable mixture, the ketchup, mashed potatoes and hot sauce in a bowl and mix. Check the seasoning and add more salt and pepper if necessary.
Heat the oil in an 8-inch nonstick saute pan and add the hash mixture. Using a rubber spatula, push the edges in to form a compact patty. Cook over medium heat for 30 seconds and then transfer to the oven for 10 to 15 minutes.
Invert the hash onto a warm plate and garnish with the fried eggs and some chopped chives to serve.
Substitute the prime rib with practically any leftover meat such as short ribs, flank steak, fillet, pork roast, roasted chicken or even turkey. If you don't have leftover mashed potatoes, cook 1 1/4 cups diced potatoes and, after draining them, smash half of the potatoes with the back of a spoon while they are still warm.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.