Preheat the oven to 375 degrees F.
Cut the onions in 1/2 and slice 1/4-inch thick, place on a baking sheet and toss with:
1/4 cup balsamic vinegar, 1/4 cup olive oil, 1 teaspoon salt and 1/2 teaspoon pepper.
Bake for 12 to 15 minutes, until the onions are tender.
Remove from oven and toss with 2 more tablespoons balsamic vinegar and cool to room temperature.
Whisk together the shallots, mustard, red wine vinegar, 1/2 teaspoon salt and 1/2 teaspoon pepper in a small bowl. While whisking, add 3/4 cup olive oil until emulsified.
To assemble, toss enough lettuce for 6 people with dressing, to taste. Place the lettuce on 6 plates and arrange the onions on top. Sprinkle with salt and pepper and serve.
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