Melt the butter in a saute pan. Add the carrots, celery, onion and leek and cook until soft and starting to caramelize, 25 to 30 minutes. Meanwhile, remove the fillets from either side of the fish by first cutting down the backbone and then working from head to tail. I always leave the tail attached to one of the fillets because I like the look of it. Discard the head and bones, or save them for making stock. All of this, of course, you can have done by the fishmonger.
Heat the oven to 425 degrees F/220 degrees C with a baking sheet on the rack (the hot baking sheet will ensure that the skin stays crisp).
When the vegetables are ready, pour over the wine and continue cooking until it has evaporated completely and the vegetables are sticky, 5 to 7 minutes. Sprinkle with salt and pepper. Remove from the heat, stir through the shrimp and thyme, and set aside.
Heat the oil in a nonstick frying pan (or the washed-out saute pan) large enough to accommodate the fish fillets. When sizzling hot, sear the fillets skin-side down just long enough to make the skin crisp, about 3 minutes. Remove.
Take the baking sheet from the oven with oven-mitted hands. Drizzle on a little oil. Lay on the fillet with the tail, skin-side down. Top with the stuffing. Lay on the second fillet, skin-side up. Slide three pieces of string underneath at intervals and tie around the fish. Slip the baking sheet back into the oven and bake until the fish is flaky and tender, no more than 10 minutes.
Remove from the oven, slide the fish onto a serving platter, strings and all, and serve.