Slice cabbage into quarters lengthwise. Cut out the the white core and discard. Slice each quarter across the grain into ¼” thick strips.
Place bacon in a large Dutch oven (with a tightfitting lid) over medium heat and cook, stirring occasionally, until browned and most of the fat has rendered.
Add onion and stir to coat in the bacon fat. Season with salt and freshly ground black pepper and cook until the onion softens and the edges start to brown (4 to 5 minutes)
Add the sliced cabbage and stir to coat in bacon fat. Cook until the cabbage begins to soften (about 4 minutes)
Stir in the brown sugar and mustard.
Deglaze the pan with the cider vinegar, scraping up any browned bits from the bottom of the pan with a wooden spoon. Add the chicken broth and season with a few pinches of salt and more freshly ground pepper.
Bring to a simmer, then reduce the heat to medium low and cover the pan tightly. Simmer, stirring occasionally, until the cabbage is soft and soupy and the bacon is tender, about 45 minutes. If the cabbage begins to look dry, add more broth or water.