How to Make It
Preheat the oven to 300°. In a large saucepan, heat the vegetable oil. Add the mushroom chunks and shallots and cook over moderately high heat, stirring occasionally, until well browned, about 10 minutes. Add 6 cups of water and the dried kombu and bring to a boil over high heat. Simmer the mushroom broth over moderate heat for 20 minutes. Strain the broth, discarding the solids. Measure out 1 cup of broth and reserve the rest for another use.
In a medium saucepan, heat 2 tablespoons of the olive oil. Add the diced turnips and cook over moderately high heat, stirring occasionally, until golden brown, 8 minutes. Add 2 cups of water, 1 tablespoon of the butter and 1 teaspoon of salt. Cover and bring to a boil. Reduce the heat to moderate and cook the turnips until they are very tender, about 20 minutes. Uncover and boil to reduce any liquid to a glaze. Coarsely mash the turnips with a spoon and season them with 1 teaspoon of the lemon juice.
In a medium shallow baking dish, sprinkle the fish with the lemon zest, crushed red pepper and 1/2 teaspoon of salt. Add the wine and remaining 1 tablespoon of olive oil and bake the fish for 16 to 20 minutes, until just cooked through.
In a small skillet, bring the reserved 1 cup of broth to a simmer. Remove from the heat and gradually whisk in the remaining 4 tablespoons of butter until incorporated. Whisk in 1/2 teaspoon of the lemon juice and season the sauce with salt.
Using a mandoline, slice the baby turnip paper-thin into a small bowl. Add the remaining 1 teaspoon of lemon juice and toss. Mound the turnip mash in shallow bowls and top with the halibut and sauce. Scatter the turnip slices over the halibut. Using a mandoline, thinly slice the whole mushrooms on top and serve.