Slice onions and place in large, nonreactive bowl—preferably glass or plastic.
Boil vinegar, water, salt, sweetener, bay leaf and any optional ingredients. Adjust sweetness; you want a sweet/sour/salty flavor that is balanced, and this will depend on the sweetener used.
Pour hot brine over onions and immediately seal tightly with plastic wrap.
Allow to cool completely, transfer to a jar and refrigerate. The pickled onions will be ready the next day, and will keep for several weeks, but for best results, prepare them a couple of weeks before needed.
Combine about 1 cup of drained pickled onions with a couple of tablespoons of olive oil and toss with a few handfuls of arugula. Other crisp salad greens would work, too, such as purslane or bibb lettuce. Season with a pinch of salt and grindings of fresh black pepper.