Egg-Salad-and-Caramelized-Onion Sandwiches

In a large skillet over medium-low heat, heat olive oil. Add sweet onions and cook, stirring occasionally, until soft and translucent, about 40 minutes. Increase heat to medium and cook, stirring, until onions begin to turn golden brown, about 15 minutes. Stir in vinegar, 3/4 teaspoon salt, and pepper, and cook, stirring occasionally, until onions are deep brown, about 10 more minutes. Transfer onions to a plate and set aside to cool. Lightly toast bread. In a large bowl, combine chopped eggs, celery, yellow onion, mustard, and remaining salt. Add mayonnaise and gently stir to mix. Season with pepper to taste. On each of 6 slices of bread, place a couple of lettuce leaves, then the egg salad and reserved caramelized onions, both divided evenly. Top each with one of the remaining bread slices, and cut each sandwich in half with a serrated knife. Serve immediately.