Build a charcoal fire or heat a gas grill.
Place the ground beef in a large bowl and add the mustard, olive oil, salt, and pepper. Mix gently with a fork to combine, taking care not to compress the ingredients. Shape the meat into 6 (3 1/2-inch) patties of equal size and thickness.
When the grill is medium-hot, brush the grill grate with oil to keep the burgers from sticking. Place the burgers on the grill and cook for 4 minutes. Using a big spatula, turn the burgers and cook for another 3 to 4 minutes, until medium-rare or more, or cook longer if you prefer hamburgers more well done.
Meanwhile, break apart the English muffins and toast the 6 halves cut side down on the grill. Spread each half with mayonnaise and top with a burger and then with a heaping tablespoon of caramelized onions. Serve hot.
Heat the olive oil and butter in a large shallow pot, add the onions and thyme, and toss with the oil. Place the lid on top and cook over medium-low heat for about 10 minutes to sweat the onions. Remove the lid and continue to cook over medium-low heat, stirring occasionally, for 25 to 30 minutes, until the onions are caramelized and golden brown. If the onions are cooking too fast, lower the heat. Add the vinegar, salt, and pepper and cook for 2 more minutes, scraping the brown bits from the pan. Season to taste (they should be very highly seasoned).
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