In a medium saucepan, combine chicken stock, 3/4 cup water, and 3 tablespoons butter. Add onion, carrot, celery, and potatoes; cook over medium-low heat until tender and most liquid is absorbed. Season with salt.
In an 8- to 10-inch skillet, bring 1/2 cup water and remaining 1 tablespoon butter to a simmer. Season halibut on both sides with salt. Place halibut in poaching liquid; cook until just well done, 3 to 5 minutes. Using a slotted spatula, transfer halibut to plate, and serve with vegetables.