Halve the cucumber lengthwise and use a spoon to scrape out the seeds, then slice the flesh crosswise into ¼ in (5 mm) pieces. Heat the butter in a saucepan until melted, toss in the cucumber, and let it cook gently for 2 minutes.
Transfer two-thirds of the softened cucumber to a bowl and mix with the chopped dill. Add salt and pepper to taste, then drizzle with lemon juice. Fill each trout cavity with this seasoned mixture.
Cover a broiler pan with aluminum foil. Lay the stuffed trout on the prepared foil and distribute the leftover cucumber pieces around them.
Place the trout under a preheated hot broiler, keeping them 4 in (10 cm) below the heat. Cook for 4–7 minutes on each side, flipping once, until the fish flakes effortlessly when tested with a fork.
Decorate the trout with fresh dill sprigs just before serving and bring them to the table while still hot. Pair with buttered new potatoes and offer the dill cream sauce on the side.