Heat 2 teaspoons oil in a large flat-bottomed carbon-steel wok or large, heavy skillet over high heat. Add eggs and cook, without stirring, until fully cooked on one side, about 30 seconds. Flip and cook until just cooked through, about 15 seconds. Transfer to a cutting board and cut into 1/2-inch pieces.

Add 2 teaspoons oil to the pan; heat over high heat. Add cauliflower in an even layer; cook, undisturbed, until lightly browned, 3 to 4 minutes. Transfer to a plate.

Add 2 teaspoons oil to the pan; heat over high heat. Add chicken breast; cook until brown, about 3 minutes. Transfer to the plate with the cauliflower.

Add the remaining 2 tablespoons oil to the pan; heat over high heat. Add broccoli, bell pepper and garlic; cook, stirring occasionally, until lightly charred, 4 to 5 minutes. Stir in soy sauce (or tamari), water, vinegar and pepper. Bring to a boil; boil for 30 seconds. Remove from the heat. Stir in the reserved eggs, cauliflower and chicken breast.