Place a heavy, nonreactive 4-quart saucepan over medium-high heat. Add the butter and garlic and cook just until the garlic starts to brown, about seven minutes, stirring almost constantly. Stir in the chopped onions, celery, and sweet pepper. Cook until the onions are clear, about five minutes, stirring occasionally. Add the tomatoes, fumet, parsley, thyme, 1 tablespoon of kosher salt and ½ teaspoon of pepper. Simmer 20 minutes.

Season the redfish pieces lightly, using a total of ½ teaspoon of salt and ¼ teaspoon pepper.

After the vegetables have simmered 20 minutes, add the fish to the liquid. Let the court-bouillon return to a simmer, and continue simmering about seven minutes more, until the fish pieces are just cooked through. To test for doneness, transfer one piece of the fish to a plate and insert the tip of a thin-bladed knife into the thickest part of the fillet for approximately ten seconds. Remove the knife and lay the tip of the blade flat against the inside of your wrist. If the tip feels hot against your skin, the fish is done.

Serve the court-bouillon warm, alongside boiled rice or pasta, in heated, shallow soup bowls, with bread for sopping up the sauce.