Add the chickpeas to a large bowl. Use a paper towel to remove any excess water. Mix in the za'atar, chile paste, and season with salt to taste. Set aside.
Heat a large pan over medium heat and add 2 tablespoons of olive oil. Add half the chickpeas and toast, stirring often, for 2 to 3 minutes. Shape them into two circles with small empty pockets in the center.
Crack two eggs into the pan, aiming the yolk for the center of the chickpea circles. Tilt the pan, letting any spicy oil pool to the side. Baste the egg whites with this oil for 30 to 60 seconds, until cooked through.
Remove the eggs from the pan, scooping out any straggler chickpeas. Repeat with the remaining olive oil, chickpeas, and eggs.
Garnish with black pepper, dill, and mint. Eat immediately.
