Chop the carrots, celery, onion, leek, and garlic.
In a large pot or Dutch oven, heat the olive oil and butter over medium heat. Add the chopped carrots, celery, onion, and leek, stirring occasionally for 3-4 minutes until softened. Add the minced garlic and stir for another minute until fragrant.
If using sourdough discard, add it now and stir for 2 minutes until fully incorporated. If using a roux, melt 2 tbsp butter, whisk in 2 tbsp flour, and cook for 1-2 minutes before proceeding.
Pour in the chicken stock and water, stirring well. Bring to a gentle boil, then reduce the heat to low and partially cover the pot. Let the soup simmer for 10 minutes to develop flavour.
Increase the heat to medium and add the small pasta, stirring occasionally. Cook until the pasta is tender, following package instructions for timing. Stir in the shredded cooked chicken and let it warm through for 1 minute.
Turn off the heat and stir in the heavy cream for a rich, velvety texture. Season with salt and black pepper to taste.
Garnish with freshly chopped dill and serve warm. Pair with sourdough bread!