Pull the meat from the bone, then discard the bones and any connective tissue.
Meanwhile, cook the shallots in the vinegar until the pan is dry.
Transfer the meat to the bowl of a stand mixer with the paddle attachment, add the thyme and shallots, then slowly work the meat, slowly drizzling in the fat until it's completely absorbed.
Taste the mixture, it should be well seasoned, but if you think it could use a little more of something, add it.
The rillettes should be smooth and spreadable, but they will firm up when chilled. They'll last in the fridge for 3-4 days in the fridge, but can be stored for months if they're packed into a container and covered with a layer of lard to act as a seal, such as in a mason jar. Serve at room temperature with crusty bread, mustard and pickles, or as part of a charcuterie plate.
Notes
For the best results, make sure the rillettes are at room temperature so the fat is spreadable.