Bring a large pot of water to boil. Cook the Brussels sprouts 2 minutes, then drain well, cool in cold water and drain well again. Place the apple juice, diced apple and lemon juice in a wide skillet and bring to a boil. Add the Brussels sprouts and lower heat to a simmer; cook until the Brussels sprouts are heated through and the apple juice is almost reduced to a glaze. Mix in the nuts, salt, pepper and butter. Spoon into a dish and serve.