2) When the grill is medium-hot, brush 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.
3) Cut the muffins in half 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 heaped tbsp of caramelized onions. Serve hot.
For the caramelized onions:
1) 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.
2) 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.
3) Add the vinegar, salt and pepper and cook for 2 more minutes, scraping the brown bits from the pan. Season to taste (the onions should be very highly seasoned).