This recipe courtesy of David Pasternack.
In a medium saucepan, heat oil over medium-low heat. Add garlic, and cook until translucent, about 2 minutes. Stir in onion, carrot, and celery, and cook until softened, stirring occasionally, about 2 minutes. Add tomatoes along with their juice, breaking them up with a whisk, and basil stems. Let simmer, stirring occasionally, until sauce has thickened, about 45 minutes.
Season with salt and pepper. Stir in chopped basil; remove from heat. May be used immediately, or let cool until room temperature and stored in an airtight container in the refrigerator for up to 3 days.