1. Combine cucumber, mustard, vinegar, horseradish, 1/8 teaspoon salt, and 1/8 teaspoon pepper in a bowl. Let stand 10 minutes, tossing occasionally.
2. Combine remaining 1/8 teaspoon salt, remaining 1/4 teaspoon pepper, dill, eggs, pastrami, and onions in a bowl, stirring with a whisk.
3. Heat a medium nonstick skillet over medium heat. Add oil to pan; swirl to coat. Add egg mixture to pan; cook 2 minutes. Cover, reduce heat to low, and cook 8 minutes or until set. Cut into 4 wedges. Place 1 lettuce leaf on each bread slice; top with 1 omelet piece and 1/4 cup cucumber mixture.