Start by finely dicing the carrots, onions, and celery, as well as chopping your fresh herbs if using.
Place the diced tomatoes (with their juice), carrots, onions, celery, oregano, basil, and chicken broth into the slow cooker. Stir gently to combine all these fresh ingredients.
Cut the butter into smaller chunks and scatter over the soup mixture. Sprinkle the flour evenly on top. Do not stir yet. Cover and cook on low for 6 hours or on high for 3 hours, allowing the butter to melt and the flour to thicken the soup gradually.
After cooking time, stir the soup thoroughly to mix in the roux. Stir in the Parmesan cheese and half and half until the cheese melts and the soup turns creamy and smooth.
Season with salt and pepper to taste. If the soup is too thick, add a splash more broth or half and half until you reach your desired consistency.
Ladle the soup into bowls and garnish with extra fresh basil or a sprinkle of Parmesan if desired. Pair it with crusty bread or a simple salad for a complete meal.