Heat oil in a large pot over medium heat; sauté onion, carrots, and celery until softened, about 6 minutes.
Add garlic and cook 30 seconds until fragrant. Stir in chicken pieces and brown lightly, 3–4 minutes.
Pour in broth and tomatoes, add thyme and bay leaf, then bring to a simmer.
Reduce heat and simmer 12–15 minutes until chicken is cooked through. Add beans and kale; simmer 5 more minutes until kale wilts.
If using pasta, add cooked pasta or cooked rice at the end. Season with salt and pepper and remove bay leaf.