I'll admit that I use water most of the time, but when you have vegetable broth on hand, it's a great nutritional addition. I recommend a lightly flavored broth so that the soup maintains the flavor of the onions and balsamic vinegar. The balsamic vinegar I use has no added sugar, which is good for sugar-free diets such as SCD.I tend to make this a bit on the sweet side and a bit on the salty side too. To bring the carbs down replace the honey with low-carb sweetener.I sometimes add up to 2 tablespoons of honey depending on the sweetness of the onions, so if you like your broth a bit sweeter feel free to add some more honey in step 4 after tasting the broth. Also add a bit more salt in step 4 if needed.
Add the oil and onions to the pot, cover, and cook on a low heat for 15 minutes.
Take the top off the pot. Raise the heat slightly and add ½ teaspoon salt, garlic and a drizzle of honey. Cook uncovered for 15 minutes, occasionally stirring the onions. The onions should be browned but not burnt. Add a bit more oil if the onions are sticking to the bottom of the pot.
Add the broth or water, 1 ½ teaspoons of salt, and balsamic vinegar, and simmer for 15 minutes. Feel free to add a bit more salt if needed.
Place soup servings in oven-safe soup bowls and sprinkle with Parmesan or Swiss cheese on top.
Place the soup bowls in an oven under a broiler or at a broil setting for 2 minutes or so, or until the cheese is bubbling.
Serve hot.
Store leftover soup in the refrigerator for a few days, or freeze for a few months.