Heat 2 tablespoons of the oil in a large nonstick skillet. Season the fish with ½ teaspoon salt and ¼ teaspoon pepper.
Cook over medium-high heat until opaque throughout, 2 to 3 minutes per side. Transfer to a plate and break into pieces.
Whisk together the lemon juice, mustard, the remaining ⅓ cup of oil, ½ teaspoon salt, and ¼ teaspoon pepper in a small bowl.
Arrange the fish, eggs, olives, anchovies, potatoes, and green beans on a plate. Drizzle with the dressing.