Using kitchen shears, cut the chilies into ½-inch pieces. Heat a medium, heavy saucepan over medium heat on the stove. Add the chile pieces and toast until fragrant and slightly darker, 4 to 5 minutes, stirring often. Add the broth, garlic, oregano, cumin, cloves, and tomato paste. Bring to a boil over high heat, stirring occasionally. Reduce the heat to medium-low, cover, and simmer until the chiles are soft, about 20 minutes. Remove from the heat and allow to cool. Transfer the mixture to a blender. Add 1 tablespoon oil, the vinegar, 1 teaspoon salt, and ¾ teaspoon pepper. Holding down the cover firmly, blend the chile mixture until the sauce is a smooth puree, but still on the thick side, scraping down the sides as needed. Pour the sauce into a small saucepan and season with cayenne pepper, more vinegar, and more salt and pepper, if desired.
Prepare the grill for direct cooking over medium-high heat (400° to 500°F).
Prepare the grill for cooking over medium-high heat (450°F)..
In a medium bowl whisk 2 tablespoons oil, the lime juice, and a pinch of salt and pepper. Mix in the tomatoes and onion. In another medium bowl gently combine the ground chuck, 1 teaspoon salt, and ½ teaspoon pepper and shape into six patties, each about ¾ inch thick. With your thumb or the back of a spoon, make a shallow indentation about 1 inch wide in the center of the patties to prevent them from forming a dome as they cook.
Brush the cooking grates clean. Grill the patties over direct medium-high heat, with the lid closed, until cooked to medium doneness (160ºF), 8 to 10 minutes, turning once. During the last minute of grilling time, place the cheese on the patties to melt, and grill the buns, cut side down, over direct heat. Remove from the grill.
Brush the cooking grates clean. Grill the patties over medium-high heat, with the lid closed, until cooked to well done (160ºF), 8 to 10 minutes, turning once. During the last minute of grilling time, place the cheese on the patties to melt, and grill the buns, cut side down, over medium-high heat. Remove from the grill.
Reheat the sauce over medium heat on the stove, stirring occasionally.
Arrange one bun half, cut side up, on each plate and top with a patty. Spoon some warm sauce over each. Mix the lettuce into the salsa and spoon over the top. Serve with any remaining sauce and salsa.