Microwave the potatoes with 2 tbsp water in a covered bowl until just tender at the edges, 4 to 5 minutes, then drain and dry them.
Heat the oil in a 12-inch cast-iron skillet over medium-high heat and spread the potatoes in one layer.
Cook without moving for 4 minutes, then turn and continue cooking until browned on several sides, about 8 minutes more.
Add the onion, ham, paprika and pepper and cook until the onion softens and the ham edges curl, 5 to 6 minutes.
Make four wells in the hash, crack an egg into each and pour the water around the edge of the skillet.
Cover and cook over medium-low heat until the whites are set but the yolks still give when touched, 4 to 6 minutes.
Scatter the scallions over the skillet and serve from the pan.