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.