Preparation

Make the sauce:

• Set a dry 10-inch skillet over medium heat for 2 minutes. Toast the ancho and pasilla chiles in the skillet for about 20 seconds on each side; don’t let them scorch. Remove the stems, seeds, and ribs from the chiles. Soak the chiles in a bowl of hot water for about 20 minutes; drain them and put them in a blender. Add the chipotle.
• Heat 1 Tbs. of the oil in the 10-inch skillet over medium heat. Add the onions and cook, stirring frequently, until soft and golden brown, lowering the heat as necessary to prevent scorching, about 10 minutes. Add the garlic and cook for 1 minute. Put the onions and garlic in the blender, along with the cilantro, brandy, and 1/4 cup water. Blend to a smooth paste, adding additional water as necessary, 1 Tbs. at a time, to puree the ingredients. Transfer the chile paste to a small bowl.
• Heat the remaining 2 Tbs. olive oil in a small saucepan over medium-high heat. When the oil is just beginning to smoke, add the chile paste. Cook, stirring constantly to incorporate it into the oil, until it’s very thick, 2 to 4 minutes; reduce the heat if necessary to prevent burning. Reduce the heat to medium and gradually stir in the broth. Add the brown sugar and salt. Simmer until the mixture is the consistency of a medium-thick sauce, 1 to 2 minutes. Season to taste with salt.

Cook the steaks:

• Position a rack 4 inches from the broiler element and heat the broiler on high. Drizzle both sides of the steaks with the lime juice and season all over with salt and pepper. Heat an 11- or 12-inch skillet, preferably cast iron, over medium-high heat, add the olive oil, and sear two of the steaks on one side, about 2 minutes. Turn the steaks and sear them on the other side, and then continue cooking, lowering the heat as needed, until they’re done to your liking, about 2 minutes on the second side for medium rare. Transfer the steaks to a rimmed baking sheet and repeat with the remaining two steaks.
• When all the steaks are cooked, turn the heat to medium, pour the chile sauce into the skillet, and stir to incorporate any browned bits and juices from the meat. Sprinkle some of the Oaxaca (or mozzarella) cheese on each steak, spoon some sauce over them, and then top them with some of the cotija or anejo (or feta) cheese. Put the baking sheet under the broiler to melt the cheese, about 1 minute, and serve immediately.