Position racks in the upper and lower thirds of the oven; preheat to 450 degrees F. Brush 2 rimmed baking sheets with olive oil. Pat the fish dry with paper towels and transfer to the baking sheets; open so the fish is skin-side down. Drizzle half of the lemon juice over the fish and season with salt and pepper. Roast, switching the pans halfway through, until the fish is cooked through and flakes easily with a fork, about 10 minutes.
Meanwhile, combine the arugula, radishes, celery, hearts of palm, parsley and Peppadews in a large bowl. Add the olive oil, lemon zest, the remaining lemon juice and the Peppadew brine; season with salt and pepper and toss to coat. Top each fish with the salad, chives and croutons.
Order whole trout from a fish counter and ask for it to be cleaned and trimmed.