Rinse and peel potatoes; cut into 1/2-inch chunks.
In a 2- to 3-quart pan over high heat, combine potatoes, onion, broth, and red peppers. Bring to a boil, reduce heat, cover, and simmer until potatoes mash easily, about 20 minutes.
In a blender or food processor, whirl potato mixture until smooth. Return to pan and add half-and-half. Quick-chill soup (nest soup-filled pan in ice water and stir often until soup is cool, 15 to 20 minutes), or cover and refrigerate until cold, at least 1 hour or up to 1 day.
Meanwhile, with a sharp knife, cut peel and white membrane from oranges. Working over a bowl, cut between inner membranes and fruit to release segments into bowl; discard peel and membrane.
Shell egg and shred or finely chop. Break trout into flakes.
Ladle soup into wide bowls. Garnish equally with orange segments and trout; sprinkle egg and chives on top. Add salt and pepper to taste.