Make the salad topping. In a medium bowl, combine the tomato, cucumber, avocado, and radishes. Add the parsley and za’atar, and season with a big pinch each of kosher salt and black pepper. Add the lemon juice and a drizzle of olive oil. Toss to combine.
Cook the eggs. In a nonstick skillet, heat about 1 tablespoon olive oil over medium heat. Crack the eggs into the oil and scramble. Just before they are set, add 2 to 3 tablespoons plain Greek yogurt and stir until the eggs are cooked through. Season with a pinch each of salt and pepper.
Assemble the breakfast hummus bowls. In one large bowl or 4 individual bowls, spread the hummus and top it with the scrambled eggs. Spoon the salad on top. If you like, finish with another sprinkle of za’atar and a little olive oil. Serve with warm pita bread.