Fennel, lemon, herbs, and white wine create a flavorful broth, often referred to as court-bouillon, for poaching halibut. To maintain a clear cooking liquid, the aromatics are bundled between a halved leek.
Place potatoes in a saucepan, and add enough cold water to cover by about 2 inches. Add 1 teaspoon salt, and bring to a boil. Reduce heat, and simmer briskly until just tender, about 15 minutes. Drain, and return to pan to keep warm.
Meanwhile, in a pot wide enough to accommodate halibut fillets in a single layer, combine 10 cups water, the wine, fennel wedges, lemon slices, and remaining 2 3/4 teaspoons salt.
Place parsley, bay leaf, and some of the reserved fennel fronds between the leek halves, and tie together with kitchen twine. Add to pot, and bring to a boil. Reduce heat, and simmer briskly until fennel is barely tender, 12 to 15 minutes.
Clip a candy thermometer to side of pan. Reduce heat so that liquid is barely simmering (190 degrees to 200 degrees). Add halibut in a single layer. Cook, adjusting heat so that steam rises but only a stray bubble surfaces, until fish begins to flake when gently pressed and an instant-read thermometer inserted into center of fillets registers 140 degrees, about 6 minutes.
While fish is cooking, slice potatoes 1/4 inch thick, and divide among 4 shallow dishes. Remove 6 lemon slices from pot, and finely dice the peel. Melt butter in a small saucepan, and stir in lemon juice.
As soon as the fish is cooked through, lift each fillet from pot using 2 slotted spatulas, and divide among 4 serving dishes. Add fennel wedges to dishes. Spoon about 1 1/2 teaspoons of the cooking liquid over each fillet; sprinkle with diced lemon peel, and drizzle with lemon sauce. Garnish with remaining fennel fronds, and serve immediately.
A slotted spatula or two eases the transfer of the finished fillets to a plate. A spoon-ful of broth can double as a finishing sauce.