Step 1

Melt butter in heavy medium skillet over medium-high heat. Add leeks and sauté until tender, about 8 minutes. Season to taste with salt and pepper. DO AHEAD Can be made 2 days ahead. Cover and refrigerate.

Step 2

Puree chives and oil in blender. Strain through fine sieve. Season oil to taste with salt and pepper. DO AHEAD Can be made 1 day ahead. Cover and refrigerate. Bring to room temperature before using.

Step 3

Bring large saucepan of salted water to boil; add thyme and bay leaf. Add potatoes and boil until tender, about 14 minutes. Drain. Cut into 1-inch pieces. Return to pan; add butter and toss to coat. Season to taste with salt and pepper. Set potatoes aside.

Step 4

Melt butter in heavy large skillet over medium-high heat. Add mushrooms and sauté until tender, about 8 minutes. Season to taste with salt and pepper. Set aside.

Step 5

Using small sharp knife, cut off peel and white pith from lemons. Working over small bowl to catch juices, cut between membranes to release segments. Add segments to juice in bowl. Bring cream to simmer in small skillet over medium heat. butter and whisk until melted. Mix in lemon segments and juice. Simmer until thick, whisking occasionally, about 2 minutes (segments will fall apart). Season to taste with salt and pepper. DO AHEAD Potatoes, mushrooms, and lemon butter sauce can be made 2 hours ahead. Let stand at room temperature.

Step 6

Heat oil in heavy large skillet over high heat. Sprinkle fish with thyme, salt, and pepper. Add fish to skillet, skin side down, and cook until skin is brown, about 5 minutes. Remove from heat. Turn fish over. Place lid on skillet to hold fish; tilt skillet to drain off excess oil. Discard oil. Add butter to skillet and tilt to melt butter and distribute evenly. Let stand covered until fish is just opaque in center, about 3 minutes.

Step 7

Rewarm leeks, potatoes, and mushrooms over medium heat and spoon onto plates; top with fish. Rewarm lemon butter sauce over medium heat, whisking frequently; spoon over fish. Drizzle with chive oil. Garnish with chervil and serve.