Heat oil in a large frying pan over medium heat. Add eggplant, onions, and garlic. Cook, stirring occasionally, until eggplant starts to soften, about 10 minutes.
Add remaining ingredients, cover, and cook, stirring occasionally, until mixture is dark and thick like chutney, about 30 minutes.
Make ahead: Up to one day; let cool, then chill airtight.
Wine pairing: Ceja 2008 Pinot Noir. The earthy berry fruit in the wine makes the sweetness of the fresh eggplant pop; a hint of sassafras doubles the effect.