Trim, peel, and thinly slice the onions, keeping the slices an even thickness.
Melt the butter in a stainless steel or enameled Dutch oven over medium-low heat. Stir in the onions, salt, pepper, and thyme, then add most of the white wine, reserving a portion for later.
Cover and cook for about 1 hour, stirring often, until the onions are very tender.
Uncover, raise the heat, and cook until the onions are caramelized and golden. Add the reserved wine, scrape up any browned bits, and cook 2 minutes more.
Stir in the broth and bring to a boil, then reduce heat and simmer uncovered for 15 minutes. Taste and adjust seasoning.
Ladle the hot soup into hot jars, packing onions to fill jars about halfway and covering with broth. Leave 1 inch headspace.
Remove air bubbles, wipe rims, set lids, and apply bands fingertip tight.
Process pints for 60 minutes or quarts for 75 minutes at 10 lbs pressure in a weighted gauge canner or 11 lbs in a dial gauge canner, adjusting for altitude (see notes).
Use a natural pressure release. Cool, check seals, label, and store.