Heat the oil in a large skillet over medium heat and add the onions, spreading them into an even layer. Cook, without stirring, until the edges of the onions are beginning to brown, 5 minutes. Stir the onions and continue to cook, stirring occasionally, until browned and caramelized, another 20 minutes. Add the chipotles, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook until fragrant and well combined, 1 to 2 minutes, and remove from the heat.
Preheat the oven broiler. Divide the meat into 12 portions, about 2 ounces each. Form a patty around each piece of cheese and flatten to 2 1/2 to 3 inches in diameter. Put the wrapped patties on a foil-lined baking sheet and sprinkle all the patties with 3/4 teaspoon salt.
Grill until the cheese is melted and the burgers are still a bit pink inside, about 6 minutes. The burgers will be cooked to at least medium in order to melt the cheese but still stay juicy.
Mix the ketchup, balsamic and 1/4 teaspoon pepper in a small bowl. Brush the burgers with some of the ketchup sauce and divide the rest among the bottom halves of the buns. Place a burger on each bottom bun, top with some of the spicy onions and finish with a top bun. Serve warm.