Melt butter in a large stockpot over medium heat. Cook and stir onion and celery in hot butter until tender, about 5 minutes.
Pour 4 cups chicken stock into the pot; add potatoes. Bring the liquid to a simmer, reduce heat to medium-low, and cook until the potatoes are beginning to soften, about 10 minutes.
Stir cod into the stock with the remaining chicken stock; season with parsley, seafood seasoning, garlic salt, salt, and black pepper. Stir flour into clam juice with a whisk until smooth; add to soup. Cook and stir until the liquid begins to thicken, 5 to 10 minutes.
Beat cornstarch into cold water with a whisk until smooth; stir into soup. Cook just long enough for the cornstarch slurry to get warm and begin to thicken the liquid even further, about 5 minutes.
Remove pot from heat. Stream half-and-half into the soup while stirring.
