Put balsamic vinegar, mustard, garlic, honey, salt, and pepper into a blender and process until smooth. With blender running, slowly pour in oil, whipping until mixture emulsifies.

Cut radicchio in half, remove core, and cut into strips. Wash and tear curly endive and arugula into bite-size pieces. Combine all greens in a large bowl with fennel.

Remove skin from trout and discard. Break trout into large bite-size pieces.

Toss salad with half of the dressing in a large bowl. Taste and add more, if needed. Divide among four plates and top with the trout pieces. Garnish with chives and serve at once.