Heat the fat over moderate heat in a large, deep sauté pan. Place the vegetable slices or halves in the pan in a single layer and sprinkle with a pinch of salt. Brown between 5 and 10 minutes (the time depends on the type of vegetable and size). Flip, salt and brown for up to 5 minutes on the other side. OPTIONAL: Remove the browned vegetables from the pan and add onions and or garlic. Sauté until softened. Return the browned vegetables to the pan.
Pour in enough liquid to create a shallow pool in the pan. Do not submerge the vegetables. The liquid should reach (at most) a third of the way up the side of the vegetables. Add a sprig or two of fresh herbs, if using. Cover. Turn the heat down to a low simmer.
Check the vegetables after about 20 minutes. They are ready when fork tender and most of the liquid has cooked off. Cabbage slices and celery need over an hour, onions cook in about 50 minutes and potatoes should be done in around 20 minutes, for example. (However, the smaller the pieces, the faster they cook.)
If the vegetables begin to dry out partway through cooking, add a splash of liquid. Remove the lid from the pan and cook off any excess liquid if necessary.