Fill a canning pot with water, cover it, and bring the water to a boil. Starting this process early is helpful because a large quantity of water may require considerable time to reach the proper temperature.

Select clean canning jars and inspect the rims carefully. Do not use jars with chips, cracks, or damaged edges that could interfere with sealing.

Clean the canning lids and screw bands with hot, soapy water. When using specialty or non-standard lids, prepare them according to the manufacturer's directions.

Pour the extra-virgin olive oil into a large heavy-bottomed saucepan, Dutch oven, or spacious skillet, then add the finely sliced sweet yellow onions.

Cook over medium heat for approximately 10–15 minutes, stirring occasionally rather than continuously. Allow the onions to soften gradually and develop an even golden color. Avoid increasing the heat to speed up the process, as gentle caramelization is important for developing the intended flavor.

Add the granulated sugar and balsamic vinegar directly to the caramelized onions and their cooking oil.

Continue cooking over medium heat for approximately 45 minutes. Stir from time to time as the liquid reduces and the mixture becomes thick, glossy, and syrup-like.

Toward the end of the cooking period, stir in the sea salt, freshly ground black pepper, and finely chopped summer savory. If summer savory is unavailable, use a blend of thyme and rosemary with a similar herbal character.

Taste the finished mixture and adjust the existing seasonings only if necessary, keeping the original flavor profile intact.

While the onion jam is still hot, ladle it into the prepared jars. A canning funnel may be used for easier filling. Leave approximately 1/4 inch of headspace between the jam and the top of each jar.

Carefully wipe the jar rims clean, position the lids, and secure the screw bands until they are finger-tight.

Using a jar lifter, lower the filled jars into the boiling-water canner. Make certain the jars are covered by at least 1 inch of water.

Return the water to a simmer and process the jars for 10 minutes. For higher elevations, add 1 additional minute of processing time for every 1,000 feet of altitude as needed.

Lift the jars from the canner and place them upright on a towel-covered surface. Leave them undisturbed for 24 hours to cool completely.

After the 24-hour cooling period, inspect each jar to confirm that it has sealed. A properly sealed lid should be drawn firmly downward and should not flex when pressed.

Remove the screw bands and clean the exterior surfaces of the jars before storing them in a cool, dry location.

Any jar that has failed to seal should be refrigerated and used before the properly sealed jars.

When the jam is not being canned, allow it to cool before transferring it to airtight containers and storing it in the refrigerator.