Heat oil in a large skillet over medium-high heat. Reduce heat to medium and add potatoes, onion, bell pepper, paprika and 1/4 teaspoon each salt and pepper. Stir to combine, then spread in an even layer in the pan and cook, undisturbed, for 3 minutes. Stir in pastrami, spread back into an even layer and cook undisturbed again until a crust starts to form on the bottom, about 5 minutes more. Remove from the heat and cover to keep warm.
Just after you add the pastrami to the hash, coat a large nonstick skillet with cooking spray and place over medium heat. Crack each egg into a small bowl and slip them, one at a time, into the skillet, taking care not to break the yolks. Season with the remaining pinch of salt and pepper. Reduce heat to medium-low, cover and cook undisturbed for about 2 minutes for soft-set yolks.
Divide the hash among 4 plates and top each serving with an egg.