How to Make It
Bring a large pot of water to a boil. Drop the potatoes in and cook until just fork tender, about 6 to 7 minutes. Drain and return the potatoes to the pot to dry out in the residual heat.
In a large skillet, heat the olive oil over medium-high heat. Add in the potatoes, salt, paprika, and fresh black pepper to taste. Toss to coat the potatoes in the oil and spices. Let the potatoes cook, untouched, for at least 4 minutes. Toss and then continue cooking, undisturbed, for an additional 4 minutes. This will allow a crust to form on the potatoes.
Once the potatoes are crispy, add in the pastrami and sauerkraut. Toss to mix with the potatoes. Sprinkle the shredded cheese over top.
Make pockets in the potatoes for the eggs to sit in and crack the eggs into the pockets.
Immediately cover and allow to cook for 2 to 2 ½ minutes until the whites are just set and the yolks are still runny.
Season with salt and fresh pepper.