Heat oil in a Dutch oven over medium heat. Add onion and salt and cook, stirring occasionally, until starting to brown, about 5 minutes. Add figs, water, honey (or brown sugar) and aniseed (or star anise). Bring to a simmer and cook, adjusting the heat as necessary to maintain a simmer, until the figs are starting to fall apart, 2 to 2 1/4 hours.Advertisement
Let cool. Remove star anise, if using, and spoon the compote into airtight containers; refrigerate.
Make Ahead Tip: Refrigerate in airtight containers for up to 3 weeks.