Heat a Dutch oven or stockpot with a lid over medium heat and add the butter. Once the butter is melted, add the leeks, salt, pepper and cayenne, and saute until the leeks are soft (10 to 15 minutes). Add the garlic and saute for about two minutes until it is lightly browned and fragrant.
Stir in the cornstarch until it is completely soaked up in the butter, and then pour in the chicken broth, stirring well.
Add the potatoes and turn the burner to high. Once the soup is boiling, reduce the heat to medium-low and pour in the cream. Place the lid on the pot and let the soup simmer for about 30 minutes or until the potatoes are fork tender. Stir it occasionally while it simmers.
Taste the soup to see if it is seasoned to your liking. If you want it a bit thicker, use an immersion blender or regular blender to partially blend the soup, making sure to leave some potato chunks.
Serve with oyster crackers and some crusty bread on the side.