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Preparation Crack eggs into a medium bowl and add 3/4 tsp. salt. Whisk until no streaks remain.
Heat 2 Tbsp. butter in a large nonstick skillet over medium. As soon as foaming subsides, add 2 slices of bread and cook until golden brown underneath, about 3 minutes. Transfer to plates, cooked side up. Repeat with another 2 Tbsp. butter and remaining 2 slices of bread. Season toast with salt. Wipe out skillet and let it cool 3 minutes.
Heat remaining 2 Tbsp. butter in reserved skillet over medium-low. Once butter is foaming, cook egg mixture, stirring with a heatproof rubber spatula in broad sweeping motions, until some curds begin to form but eggs are still runny, about 2 minutes. Stir in crème fraîche and cook, stirring occasionally, until eggs are barely set, about 1 minute.
Spoon eggs over toast and top with trout. Finely grate lemon zest from one of the lemon halves over trout, then squeeze juice over toast. Season with pepper; scatter scallions and dill on top.
Squeeze juice from remaining lemon half into a medium bowl. Add arugula and drizzle with oil; season with salt and pepper. Toss to coat. Mound alongside toasts.