Heat a large nonstick skillet over medium-low heat until hot. Add oil. When oil is hot enough to sizzle a piece of potato, spread potatoes in pan. Cook potatoes, turning occasionally with a wide spatula, until they are evenly browned and crisp, 10 to 15 minutes. Add onion and cook, stirring, for 3 minutes. Sprinkle potatoes with salt and paprika.
Meanwhile, bring medium saucepan half full of water to a boil. Add broccoli and cook, uncovered, for 3 minutes. Drain, rinse with cold water and shake dry. Add broccoli and peppers to skillet, distributing them evenly on top of potatoes.
In small bowl, whisk the eggs, 2 tablespoons water and a grinding of black pepper until frothy. Pour eggs over vegetables. Sprinkle cheese on top.
Cook over medium to medium-low heat, tilting pan to allow uncooked eggs in the center to run under the set eggs around the edges, for 5 minutes. Cover and cook until eggs are evenly set, 3 to 5 minutes more. Cut into wedges and serve directly from the pan.