1. Preheat oven to 275 degrees. Season ribs generously with salt and pepper. In an 8-quart Dutch oven or heavy pot, heat oil on high. In batches, brown ribs on all sides, about 20 minutes total. If oil gets too dark during browning, pour off and add more oil (do not wipe pot clean). Transfer ribs to a plate and pour off all but 2 tablespoons fat from pot.

2. Add onions, celery, carrots, garlic, thyme, bay leaves, rosemary, cumin, and star anise. Cook, scraping up any browned bits, until onions soften, 5 minutes. Return ribs to pot and add broth and root beer. Bring to a rapid simmer, cover, and place pot in oven.

3. Cook until meat can be easily pierced with the tip of a paring knife, about 3 hours. (To store, refrigerate, up to 4 days.) With a slotted spoon, transfer ribs to a large straight-sided skillet. Skim off and discard any fat from cooking liquid. Strain through a fine-mesh sieve into skillet; discard solids. Boil over medium-high until liquid is reduced by half, about 20 minutes. Continue to cook, spooning liquid over meat occasionally, until ribs are glazed and sauce is thickened, 5 to 7 minutes. Season with salt and pepper.