Preheat oven to 400 degrees F. Toss the potato halves in 2 tablespoons olive oil, paprika powder, salt and pepper. Stir well.
Spread the potato cubes on a large baking sheet, lined with parchment to prevent them from sticking. Bake the potatoes in the preheated oven until they are cooked through, about 30 minutes.
Meanwhile, heat 2 tablespoons vegetable oil in a skillet over medium-high heat. Add the sliced mushrooms and fry uncovered until mushrooms are golden and all liquid is absorbed. Reduce heat to medium. Add the green onion and cook for 5 minutes. Add the potatoes, stir and adjust the seasonings.
Heat enough oil to lightly coat the bottom of a small non-stick skillet over medium heat. Crack the eggs in single layer into the skillet. Season with salt and pepper. Cook until desired doneness (a runny yolk will be perfect for this recipe).
Divide the potato-mushroom mixture evenly in the serving dishes. Place the egg on top and sprinkle with cilantro.