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Melt the butter in a large pot with a lid. Add the leeks, celery, and garlic and cook, stirring occasionally, until the vegetables are tender, 5 to 7 minutes. Add the potatoes, 1 ½ teaspoons salt and ½ teaspoon pepper.
In a small bowl, whisk 1 cup of the chicken stock and the cornstarch together until smooth. Add to the soup along with the remaining 3 cups stock. Bring to a simmer, partially cover the pot, and cook until the potatoes are very tender, about 20 minutes. Puree the soup with an immersion blender. Taste, and season with more salt and pepper as needed.