In a large bowl, combine the chicken, lemon zest, 1/2 tablespoon lemon juice, 1/2 tablespoon olive oil, 1/2 teaspoon oregano, salt, and a few grinds of pepper. Marinate for about 10 minutes.
Preheat a grill pan, skillet, or wok over medium-high heat. Add 1/2 tablespoon olive oil and then the chicken. Grill or stir-fry until the chicken is cooked through, approximately 8 minutes. Transfer the chicken to a paper towel-lined plate.
In a 1-cup measuring cup or small bowl, combine the yogurt, remaining 1/2 tablespoon lemon juice, remaining 1 tablespoon olive oil, remaining 1/2 teaspoon oregano, and a few grinds of pepper. If the dressing is too thick for your liking, dilute it with a tablespoon or so of water.
In a large bowl, toss the lettuce with a few spoonfuls of the dressing. Divide between two serving bowls and top with chicken, tomatoes, cucumbers, and feta cheese. Drizzle with additional dressing and serve with pita chips, if desired.