Toss chard stems, shallot, and salt in a colander set in the sink.
Let stand 1 hour. Rinse and drain well.
Meanwhile, toast mustard and caraway seeds in a small skillet over medium heat, stirring often, until mustard seeds begin to pop, about 2 minutes.
Let seeds cool.
Bring vinegar, sugar, and 1 cup water to a boil in a small saucepan; let cool slightly.
Pour brine into jars.
Let cool slightly, then cover and chill.
DO AHEAD: Pickles can be made 2 weeks ahead. Keep chilled.
Pickled Green Papaya
Use the same salting method and brine, but instead of chard stems and shallot, use 1 sliced seeded peeled green papaya (about 1 lb.). Omit mustard and caraway seeds, and use 2 red or green Thai chiles and one 1"-piece peeled turmeric or ginger, thinly sliced.Pickled Banana Chiles
Use the same salting method and brine, but instead of chard stems and shallot, use 1 lb. fresh banana chiles (cut a long slit in each pepper to allow brine to penetrate) and 1 sliced peeled carrot. Omit mustard and caraway seeds, and use 1 garlic clove.