Preheat the broiler. Toss the chicken with 1 tablespoon olive oil, the juice of 1 lemon, the garlic, mint, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl. Set aside 10 minutes.
Meanwhile, toss the pita wedges with 1 tablespoon olive oil on a baking sheet and season with salt and pepper. Broil until lightly golden, about 1 minute per side; transfer to a large bowl.
Arrange the chicken on the baking sheet. Broil until cooked through and golden, about 4 minutes per side. Let cool.
Tear the pita pieces in half. Add the cucumbers, tomatoes, parsley and scallions to the bowl, then add the remaining 2 tablespoons olive oil and the juice of the remaining lemon; toss. Chop the chicken and add to the bowl along with the greens. Season with salt and pepper and toss. Serve topped with the feta; drizzle with olive oil.