Melt the butter in a stainless steel or enameled Dutch oven, cooking over medium-low heat.
Stir in the onion, along with the salt, black pepper, and thyme, along with 2 cups of the white wine.
Cover the mixture and cook for an hour, or until the onion is tender. Stir often to prevent burning.
Toward the end of this cooking time, prepare a pressure canner by adding a few inches of water to the bottom and bringing it up to a simmer. Also prepare jars, lids and rings.
Uncover the pot and cook at a higher heat until the onion is caramelized. Add the rest of the white wine, then cook for an additional two minutes. Make sure you loosen any browned bits from the bottom of the Dutch oven.
Stir in the broth, then bring to a boil.
Reduce the heat, then simmer, uncovered, for 15 minutes.
Ladle the soup into hot jars. Leave an inch (2.5 cm) of headspace.
Remove air bubbles and wipe any food residue or liquid from the rims of the jars.
Place the lids on the jars, then apply bands and adjust until they are fingertip-tight.
Put the jars on the rack inside your simmering pressure canner.
Put the lid on the canner, then turn it to the locked position. Turn the heat on medium-high, then vent steam for 10 minutes.
Put the counterweight or weighted gauge on the vent, then bring the pressure to 10 lbs for a weighted gauge canner or 11 lbs for a dial-gauge canner. (Adjust canning weight for altitude, see notes)
Process pint jars for 60 minutes or quart jars for 75 minutes. Turn off the heat, then cool the canner to zero pressure.
Let the jars cool for another five minutes after depressurizing before you remove the lid.
Cool the jars in the canner for another ten to fifteen minutes, then put them out to cool at room temperature.
Label the jars, check the seals, and store.