Peel onions, slice 1/2” thick, and separate into rings. Put rings into Ziploc bag. In a small bowl, stir together balsamic vinegar, Mrs. Dash, and dried thyme. Whisk in olive oil.
Pour marinade into plastic bag and let onions marinate in refrigerator for 4 hours or more, turning a few times if able.
When ready, preheat oven to 450° F.
Meanwhile, drain onions in a colander placed in the sink. Spray non-stick baking sheet with olive oil. Arrange onions on baking sheet, in a single layer as much as possible. Roast onions for 30 mins, turning every 10 mins, or until nicely brown.
Remove onions to serving dish, sprinkle with Parmesan cheese, season to taste with salt and fresh ground black pepper. Serve hot.