In a medium saucepan over medium heat, heat the olive oil. When the oil begins to shimmer, add the onions, salt, and a few grinds of pepper. Cook, stirring occasionally, until the onions begin to brown at the corners—about 10 minutes. Add the balsamic vinegar and cook, stirring, for 1 minute to reduce slightly. Set aside to cool down.
While the onions cool, put the figs and water in a small saucepan over medium-low heat. Bring to a simmer and allow to cook, partially covered with a lid, for about 15 minutes, until most of the liquid has been absorbed.
Transfer the onion and balsamic mixture and the figs (including any liquid in the pan) to a blender or food processor and process until smooth—about 1 minute. Taste and add salt and pepper as needed. Allow to cool, then transfer to a tightly sealed airtight container.
Notes
Use 1 cup roughly diced shallots in place of the onions in this recipe for a slightly milder onion flavor.