Step 1: Place the olive oil, balsamic vinegar, and honey in a large nonreactive mixing bowl and whisk until well mixed. Season with a little salt and pepper.

Step 2: Cut each onion lengthwise into quarters, leaving the skin on and the root end intact. Add the onions to the marinade and stir to coat evenly. Let the onions marinate in the refrigerator, covered, for at least 2 hours, preferably overnight. Gently stir the onions from time to time so that they marinate evenly. You can also marinate the onions in a resealable plastic bag.

Step 3: Drain the marinade off the onions into a heavy nonreactive saucepan and set aside. Place the onion quarters in a flat rotisserie basket and close it tightly.

Step 4: When ready to cook, place the drip pan in the bottom of the rotisserie. Attach the basket to the rotisserie spit, then attach the spit to the rotisserie and turn on the motor. If your rotisserie has a temperature control, set it to 400 degrees. Cook the onions until they are dark brown and very tender, 50 minutes to 1 hour. Use a skewer to test for doneness; it should pierce an onion easily.

Step 5: Meanwhile, make the sauce: Bring the marinade to a boil over medium-high heat and let boil until thick, syrupy, and reduced by about a third, 8 to 12 minutes. Taste for seasoning, adding more salt and/or pepper as necessary.

Step 6: Transfer the onions to a serving dish, spoon the balsamic vinegar sauce over them, and serve at once.