Preheat broiler. Rinse the tomatillos under warm water to remove any stickiness and dirt. Place the tomatillos, onion and serrano on a baking sheet lined with foil and broil 3 to 4 inches from heat, turning once, until tomatillos are softened and slightly charred, about 5 to 8 minutes.
Transfer the vegetables to a blender. Add the cilantro, cumin, sugar and salt, and purée until smooth.
Heat oil in a large, nonstick skillet over medium heat. Add the salsa and cook, stirring, until slightly thickened, about 5 minutes. Add the broth and cook for another 5 minutes. The salsa should be slightly thickened.
Meanwhile, in a large, nonstick skillet, melt the butter over moderate heat. Crack the eggs one at a time into a ramekin and then slip into the skillet. Cook the eggs, sunny-side up, until the whites are firm and the yolks runny, about 3 to 4 minutes.
Gently stir the tortilla chips and chicken into the sauce, making sure they are well coated. Divide the chilaquiles among 4 plates and top each with the sour cream, cheese, red onion, cilantro and an egg. Serve with lime wedges.