Add the chicken, carrots, celery, onion, potato, tomato, tomato paste, Italian seasoning, granulated garlic, stock, 1 tablespoon balsamic vinegar, olive oil, and salt to a 6-quart slow cooker.
Cover and cook on high until the chicken and veggies are cooked through, about 4 hours on high, or 6 hours on low (times vary depending on slow cooker).
Carefully ladle out ¼ cup broth into a small bowl and whisk with the arrowroot powder, and 1 tablespoon of balsamic vinegar until smooth.
Pour the mixture back into the slow cooker, stirring to combine.
Cover and cook on high for an additional 5 minutes, until the stew begins to thicken.
Remove the chicken and shred or dice.
Add back to the slow cooker, stir, and taste, adding additional salt if desired.
Serve warm with a generous drizzle of olive oil and freshly ground pepper, if desired.
