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To make the caramelized onions:
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Place a large skillet over low heat and when it is hot, add the oil.
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Add the remaining ingredients and cook, stirring only occasionally as necessary, until the onions are deeply browned and slightly gooey, about 40 minutes.
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If the onions dry out, add water, 1 tablespoon at a time.
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Use immediately or cover and refrigerate for up to 2 days.
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(Yield: 3/4 to 1 cup)
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To make the burgers:
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Place the beef on a work surface and divide into 4 balls of equal size.
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Divide the blue cheese into 4 portions of equal size.
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Using your thumb or the back of a tablespoon, make a shallow hole in each portion of beef and fill it with the blue cheese.
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Shape the burger and carefully seal the opening.
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Form each into a patty about 3/4 to 1 inch thick, tossing it back and forth between your hands.
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To ensure more even cooking, make a 1/2-inch indentation with your thumb in the center of the burger.
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Handle the patties as little as possible; do not work more than necessary.
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15
Prepare a grill to medium-high.
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When the coals are glowing red, after 15 to 20 minutes, cover with the grate.
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After 5 minutes, use a wire brush to thoroughly clean the grate.
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Brush the grate with oil.
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When the coals are covered with pale gray ash and you can leave your hand 5 inches above the fire for 2 to 3 seconds, the coals are ready.
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Sprinkle both sides of the burgers with the salt and pepper.
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Place the burgers on the grate and grill until well seared on both sides, about 5 minutes for medium rare or 6 minutes for medium.
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Top with a few crumbles of cheese and let melt, if desired.
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Transfer to buns (or toast as shown) or a serving platter and serve immediately.
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24
Alternatively, place a cast iron skillet over high heat and when it is hot but not smoking add the burgers to the dry pan.
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25
Cook until well seared on both sides, 8 to 10 minutes for medium rare or 10 to 12 minutes for medium.