Season the fish lightly and evenly with salt. Cover loosely and refrigerate. Try not to skip this step as it really helps keep the fish nice and juicy.
Gently simmer the potatoes in salted water until fork tender and soft on the edges. Don’t worry if a few potatoes seem to be too tender as they will diffuse into the sauce.
Drain the potatoes well and place them, still steaming, in a 12 or 14-inch ovenproof skillet. Add the leeks, thyme, chicken stock, and wine. Set skillet over medium heat and stir as the broth comes to a simmer. Add 4 tablespoons of the butter and give it a gentle stir. Taste a potato to see if the dish needs more salt and pepper. Reduce the heat to low and add the 4 pieces of fish (evenly spaced in skillet).
Using a spoon or pastry brush, baste the surface of the fish with the buttery broth. Make sure no bits of leek are stuck on top of the fish or edges of the skillet. Place the skillet under the broiler. Cook until the surface of the fish and potatoes are lightly golden, about 5 to 6 minutes.
The liquid at the top of the pan should be bubbling. Turn off broiler and reduce the oven temperature to 500 degrees F. Cook for another 1 to 5 minutes until fish has an internal temperature of 130 degrees F.
While the fish is cooking, set a small, shallow platter and four plates to warm in the oven for a minute or so, then remove.
Transfer the skillet to the stove top. Using a fish slice or spatula, lift the fish, tilt a little to drain any broth, then place on the warm platter and tent with foil. Please don’t stack fish, as fish will fall apart.
Swirl the skillet gently over medium heat to encourage the potatoes to thicken the sauce as it simmers. If potatoes are being stubborn, mash a couple with a fork. Add the remaining butter and stir in.
Once the sauce has a nice body, take another taste, feel free to add a splash or two (1 teaspoon) of Champagne vinegar to add brightness.
Transfer the fish to warmed plates or bowls, and spoon the potatoes and sauce over.