Pat halibut dry with paper towels and sprinkle with ½ teaspoon salt.
Heat 1 tablespoon oil in a 12-inch skillet over medium heat until just smoking.
Place halibut skinned side down in skillet and cook about 15 seconds.
Carefully transfer halibut to a plate.
Add leeks, mustard, and ¼ teaspoon salt to oil left in skillet and cook over medium heat, stirring frequently, until softened, 8 to 12 minutes.
Stir in wine and bring to simmer.
Return halibut raw side down on top of leeks.
Reduce heat to medium-low, cover, and simmer gently until halibut flakes apart when gently prodded with paring knife and registers 140 degrees, 6 to 10 minutes.
Carefully transfer halibut to serving platter, tent loosely with aluminum foil, and let rest while finishing leeks.
Return leeks to high heat and simmer briskly until mixture is thickened slightly, 2 to 4 minutes.
Season with salt and pepper to taste.
Arrange leek mixture around halibut, drizzle with extra oil, and sprinkle with parsley.
Serve with lemon wedges.
Variations
Braised Halibut with Fennel and Tarragon
