To make the orange-honey vinaigrette, put oil, vinegar, zest, honey, salt, and black pepper in a small screw-top jar. Cover and shake well.

Stir together flour, 1/4 cup dukkah, and cayenne in a shallow dish. Beat together 2 eggs with water in another shallow dish.

Cut each chicken breast crosswise into 2 equal pieces. Pound each piece between 2 sheets of plastic wrap to 1/4- inch thickness with the flat side of a meat mallet. Sprinkle chicken with 1/4 teaspoon salt. Dip each piece in egg mixture, then dredge in flour mixture to coat.

Heat oil in a large skillet over medium heat. Add chicken; cook, turning once, until no longer pink, 4 to 6 minutes. Drain on paper towels. Wipe out skillet.

Crack remaining 4 eggs into the skillet. Sprinkle with 1/2 teaspoon dukkah and 1 pinch salt. Reduce heat to low. Cook until whites are completely set and yolks start to thicken, 3 to 4 minutes.

Arrange greens on a serving platter; drizzle vinaigrette on top. Top with chicken and eggs.