If using fresh green beans, cook in a large saucepan of boiling water until crisp-tender, about 3 minutes. Drain and refresh under cold running water.
Heat oil in a large nonstick or cast-iron skillet over medium heat until hot enough to sizzle a piece of potato. Spread potatoes in an even layer and cook, turning every few minutes with a wide spatula, until tender and browned, 15 to 20 minutes for raw potatoes, 10 to 12 minutes for cooked. Stir in the green beans, garlic, crushed red pepper, salt and pepper.
Crack each egg into a small bowl and slip them one at a time into the pan on top of the vegetables, spacing evenly. Cover and cook over medium heat until the whites are set and the yolks are cooked to your taste, 3 to 5 minutes. Sprinkle the eggs with paprika, if desired, and serve immediately.