Heat a 3-quart saucepan over high heat. Add in pork belly and turn the heat down to medium. You are doing this to allow the pork belly to gently render some fat. Sauté pork belly until the meat is cooked (about 5 minutes).
Remove all but 2 tablespoons-ish of the fat. No need to be crazy precise here, just take out any egregious fat so that your soup doesn’t taste ultra fatty afterwards.
Add shredded cabbage and mushrooms and stir well until the veggies are coated with the remaining rendered fat.
Pour in broth, miso paste, and sesame paste and stir to incorporate those into the broth. When the broth comes to boil, remove the pot from the heat. Stir in vinegar and serve.
Make sure you remove the extra fat from the saucepan before adding the rest of the ingredients. If you leave too much fat in the pan you'll end up with a fatty tasting soup, and nobody wants that
This Pork Soup With Cabbage can be refrigerated for 3-4 days. It will collect fat on top, so just simply skim the fat away or gently reheat it.
The cabbage in the soup won't handle too many reheats, so try not to reheat the whole batch if you plan on eating it throughout the week. You should only really reheat the cabbage once.
If you can't find or choose not to eat pork belly, you might want to try pork chops cut into smaller pieces.
If you don't eat pork, or just want a change of pace, you could try this with chicken.