Place celery in the bowl of a food processor; process until minced (you should have about 1 cup). Repeat process with carrots (you should have about 1 cup).
Heat oil in a large saucepan over medium-high heat. Add celery, carrot, and onion; cook, stirring occasionally, until mixture is no longer steaming and vegetables are caramelized, about 1 hour.
If not using immediately, let cool and store in an airtight container, refrigerated, up to 1 week.