Add the potatoes to a medium saucepan and cover with water. Bring the water to a boil and simmer until the potatoes are tender, 10-15 minutes.Drain the water and set aside to allow to cool.If making the day before, place cooked potatoes in an airtight container and store in the refrigerator.
Remove the steak from the refrigerator and allow to sit for at least a half hour. Then heat some of the oil in a large iron skillet over medium-high heat. Rub the remaining oil over both sides of the steak and season liberally with kosher salt and a light sprinkle of pepper on both sides. Add the steak to the skillet and cook until medium-rare, about 4-6 minutes per side depending on the thickness of your steak. Remove the steak from the skillet and allow it to rest.
Add onion, bell pepper and sliced mushrooms to skillet and cook until tender, about 5-6 minutes. Add potatoes and paprika and season with salt and pepper. Cook until potatoes begin to brown, about 5 minutes.Chop steak into 1" cubes and add to skillet. Stir and cook for a couple of minutes longer.
Using the back of a tablespoon, make 4 wells in the hash mixture. Crack an egg into each well and season each egg with salt and pepper. Cover skillet with a lid and cook until eggs are set to your liking. About 4 minutes for a runny yolk and 6-7 minutes for a set yolk.Serve with steak sauce or hot sauce for an extra layer of flavor.
Our steak and eggs hash is a great way to use leftover potatoes and steak. However, if you want to make it fresh for brunch or as a hearty weekend breakfast, you can save time by peeling, chopping and cooking the potatoes the day before. Store the cooked potatoes in an airtight container in the refrigerator.Be sure to oil and use kosher salt to liberally season the raw steak at least a half hour before cooking. Doing this will draw the moisture to the surface of the steak which will dissolve the salt and create a brine that tenderizes the steak and allows for crusty flavorful browning.