If using salt-cured capers, put in a small bowl, cover with water, and let soak 10 minutes to remove salt; rinse and pat completely dry. (For brine-packed capers, rinse well and pat dry.)
In a 4- to 6-quart pan over high heat, bring 2 to 3 quarts water to a boil. Add potatoes and cook, uncovered, until tender when pierced, about 15 minutes. Drain.
Meanwhile, pull off and discard skin from trout. Break the fish into bite-size pieces.
Set a 1- to 1 1/2-quart pan over medium-high heat. When hot, add olive oil; when it begins to ripple, add capers (use a potholder and stand back--the oil may splatter). Cook, stirring once, until berries have opened and are light brown and crisp, about 1 minute. Pour into a fine wire strainer over a 1-cup glass measure; reserve oil. Pour capers onto paper towels to drain.
In a large serving bowl, whisk oil from capers with lemon juice and pepper. Stir in potatoes. Add trout, apple, and dill and mix gently. Sprinkle fried capers over the top. Serve with sour cream, if desired.