Heat oven to 375ºF (190ºC)with rack set in middle position.
Separate pita breads into two thin rounds by peeling apart (cut edges with kitchen shears if needed).
In a large bowl, toss with 1 tablespoon olive oil then season with ½ teaspoon salt and pepper.
Arrange pita pieces smooth side down on a wire rack set inside a sheet pan.
Bake until pita pieces are crisp and golden brown, 12–15 minutes. Let cool on rack.
Meanwhile, add shallot, 2 tablespoons lemon juice, 1 teaspoon lemon zest, garlic, and ½ teaspoon salt to a now empty large bowl; toss with hands to combine then set aside.
Slice cucumber in half lengthwise. Place cut-side down on a cutting board and with heel of hand, smash down until cucumber cracks. Tear into bite-sized pieces.
Cut tomatoes into ¾-inch chunks. Transfer to a colander and let drain.
Add remaining 2 tablespoons oil and balsamic to bowl with lemon juice, shallots and garlic.
Add cucumber, tomatoes, and herbs to bowl; toss to coat then add sumac and toss to combine. Season with salt and pepper to taste.
Right before serving, stir in half of the pita bites, then transfer to a bowl and top with remaining pita bits.
Alternatively, serve vegetable salad on a large platter and spread pita chips over top.