1. In a medium pot, bring potatoes to a boil in salted water over high; reduce to a simmer and cook until tender when pierced with a knife, 10 minutes. When cool enough to handle, halve potatoes.

2. Meanwhile, heat broiler. Brush 1 tablespoon oil on one side of each bread slice. Broil, oil side up, until crisp and lightly browned. Combine cucumber, yogurt, dill, and lemon juice; season with salt and pepper. Gently stir in trout. Top each bread slice with a few lettuce leaves and trout salad.

3. In a large bowl, whisk together remaining 2 tablespoons oil, lemon zest, mustard, and 1 tablespoon hot water. Add potatoes and toss to combine. Season with salt and pepper and serve with open-faced sandwiches.