In a large stockpot, heat oil over medium-high. Add about 2/3 of the onion and cook, stirring regularly, until richly browned, 8 to 10 minutes. Add sauce and broth and bring to a rolling boil. Turn off heat and stir in 2/3 of the shredded cheese.
Place one layer of chips into a 13x9-inch baking dish and pour half of the sauce, so that all of the chips are coated. Add another layer of chips and pour remaining sauce. Sprinkle with remaining shredded cheese. Lightly cover baking dish with foil, being careful not to let the top of the chips touch the foil.
Bake 8 to 10 minutes. When chilaquiles are bubbling, remove from the oven and sprinkle with the remaining onion.
Top with queso fresco, cilantro, and sunny side up eggs, if using.
You can use green enchilada sauce / salsa verde for chilaquiles verdes.