Salsa Alchemy: In a medium bowl, combine the diced cucumber, tomato, rinsed chickpeas, crumbled feta, chopped mint, lemon juice, and black pepper. Gently stir until all ingredients are well-distributed. Set aside to allow the flavors to marinate while you prepare the eggs.
Aromatic Infusion: Place the large nonstick skillet over medium-low heat. Add the extra-virgin olive oil and sliced garlic. Warm gently for 60-90 seconds, swirling the pan, until the oil is fragrant. Remove and discard the garlic slices, leaving the infused oil.
Pita Preparation: While the oil infuses, lightly toast the whole-wheat pitas until they are warm and pliable. Once toasted, carefully slice each pita in half to create four pockets.
Egg Perfection: Increase the skillet heat to medium. Crack the eggs into the garlic-infused oil, leaving space between them. Immediately cover the skillet with a lid and cook for 2 minutes to set the whites.
Flavor Sealing: Remove the lid. Continue to cook, uncovered, for another 1-2 minutes, or until the whites are fully set but the yolks remain soft and jammy. Sprinkle the eggs evenly with the salt-free za'atar during this final stage.
Final Assembly: Carefully open each warm pita half. Use a spatula to place one za'atar-dusted egg inside each pocket. Spoon a generous amount of the Lemony Chickpea Salsa over each egg, dividing it evenly. Serve immediately.