Serves 42 tablespoons extra-virgin olive oil, plus more as needed
1 ½ pounds red onions, thinly sliced (about 4 cups)
2 tablespoons balsamic vinegar
1 teaspoon coarse salt
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1/8 teaspoon freshly ground black pepper
3 ounces firm blue-veined cheese (such as Stilton, Cashel blue or Cabrales), crumbled
1 ½ pounds ground beef
1 tablespoon finely chopped Italian parsley
1. Heat a 10-inch cast-iron skillet over medium heat. Add oil and heat until hot enough to sizzle an onion ring. Add onions, adjust heat to medium-low and sauté, stirring, for 2 minutes. Cover and cook 3 minutes or until wilted. Uncover and cook, stirring occasionally and adjusting the heat between medium and medium-low, for about 20 minutes or until onions are a dark golden brown. Add salt and pepper. Remove from heat. With a large spoon, transfer onions to a large bowl. Add vinegar, additional salt and pepper to onions and set aside. Rinse and dry pan.
2. Divide ground beef into eight 3-ounce portions, and shape each portion into a patty about 4 inches in diameter. Divide 2 ounces (about ½ cup crumbled) of the cheese evenly among 4 patties, sprinkling it on the surface and gently pressing into the meat. Top with remaining 4 patties to make 4 cheese-stuffed burgers. Press around the edge of each burger to seal top and bottom patties together. Brush both sides of burgers lightly with oil.
3. Heat skillet over medium heat until hot enough for a drop of water to sizzle on contact. Place burgers in the pan, increase heat to high, and cook, flipping them at the midpoint with a spatula, for 4 to 5 minutes per side, or until cooked to desired doneness. Transfer to a platter.
4. Add onions to the hot skillet and reheat briefly over low heat for about 1 minute, or until hot. Spoon onions over the burgers. Sprinkle with remaining ounce of cheese and the parsley, and serve.