Preheat the oven to 350 degrees. Season the fish with salt and pepper. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat until shimmering. Add the fish and sear until golden on the bottom and cooked halfway through, about 3 minutes. Flip onto a baking sheet and bake until cooked through, 3 to 5 more minutes.
Meanwhile, wipe out the skillet, return to medium-high heat and add the remaining 3 tablespoons oil. Add the mushrooms and cook, without stirring, until browned on one side, about 1 minute. Stir and cook until browned all over, about 3 more minutes. Add the shallots and cook until soft, stirring, about 2 minutes. Whisk in 2 tablespoons vinegar and the mustard and bring to a boil. Remove from the heat and stir in the chives and parsley.
Drizzle the arugula with oil and vinegar in a bowl, season with salt and pepper and toss. Divide among plates and serve with the fish. Spoon the mushrooms and pan juices on top.
Photograph by Antonis Achilleos
Recipe courtesy Food Network Magazine