Preheat a grill to medium-high. Finely chop enough of the onion to yield 2 tablespoons and transfer to a small bowl. Stir in the yogurt, dill, lemon peel, 1 tablespoon lemon juice and 1 tablespoon olive oil; season with salt and pepper.
Brush the chicken with the remaining 1 tablespoon olive oil and season with salt and pepper. Arrange on the grill, cover and cook, turning once, until golden and grill marks appear, about 8 minutes. Transfer to a plate and sprinkle the remaining lemon juice on top. Let stand for 5 minutes before thinly slicing.
In a bowl, toss the romaine, cucumber, tomatoes and remaining onion with the dressing. Season with salt and pepper; top with the sliced chicken.