In a large pot, heat 1/4 cup of the oil over medium-high heat. Add the onions and cook, stirring, until softened, about 5 minutes.
Add the greens, a handful at a time, letting each batch wilt a little bit before adding more. Once all of the greens have been added, add the broth and bring to a gentle simmer. Cover the pan and reduce the heat to low. Cook, tossing from time to time, until the greens are tender, about 20 minutes. If you like them silken and Southern, continue cooking the greens for 20 to 30 minutes longer.
Heat a separate large skillet over medium heat and fry the eggs in the remaining 2 tablespoons oil to desired doneness. Sprinkle with paprika and place over each serving of greens.