Heat 1 tbsp of the olive oil in a large lidded skillet over medium heat. Add the sweet potatoes, onion, bell pepper, smoked paprika, garlic powder, 1/2 tsp of the salt, and the black pepper; stir until everything is evenly coated.
Cover and cook for 10 minutes, stirring every 3 minutes, until the sweet potatoes are nearly tender and the onion is soft. If the pan looks dry, add a splash of water.
Uncover, stir in the spinach, and cook for 1 minute, until just wilted. Spread the hash into an even layer.
Using the back of a spoon, make 4 wells in the hash. Drizzle the remaining 1 tbsp olive oil into the wells and crack one egg into each.
Cover and cook for 5 to 7 minutes, until the egg whites are set and the yolks are still soft, or longer for firmer yolks.
Sprinkle the eggs with the remaining 1/4 tsp salt and extra black pepper, then serve hot straight from the skillet.