Preheat the oven to 350 degrees F. Toss the chicken in a large bowl with the lemon zest, 1 tablespoon each lemon juice and olive oil, 1 teaspoon oregano, 1/2 teaspoon salt and a few grinds of pepper; let sit for at least 5 minutes.
Spread the pita pieces on a baking sheet and toast in the oven, tossing halfway through, until crisp, 10 to 12 minutes. Let cool.
Meanwhile, preheat a grill pan over medium-high heat. Add the chicken and cook, turning occasionally, until cooked through, 6 to 8 minutes. Transfer to a plate.
Make the dressing: Combine the yogurt, remaining 1 tablespoon lemon juice, 2 tablespoons olive oil and 1 teaspoon oregano, 1/4 teaspoon salt and a few grinds of pepper in a small bowl. Stir in any juices from the plate of cooked chicken and 2 to 3 tablespoons water.
Toss the lettuce, a few spoonfuls of the yogurt dressing and the pita chips in a large bowl; season lightly with salt and pepper. Divide among shallow bowls. Top with the chicken, tomatoes, cucumber and feta. Drizzle with the remaining dressing.