Scramble the eggs: Whisk 3 whole eggs together in a bowl. Heat a small amount of oil in a pan and scramble until just cooked through. Remove and set aside on a plate.
Sauté the aromatics: Heat 1.5 teaspoons cooking oil (vegetable or canola) in the same pan over medium-high heat. Add 1 tablespoons minced garlic, 1 tablespoons minced fresh ginger, and the white parts of 3 green onions (scallions), sliced, white and green parts separated. Stir constantly until fragrant, about 30–45 seconds
Add cabbage and carrot: Add 0.5 medium head of cabbage, sliced thin and 1 carrot, peeled and julienned or shredded to the pan. Stir-fry over high heat until vegetables begin to soften but still have a slight bite — do not overcook.
Add the sauces: Pour in 1 tablespoons oyster sauce and 1 teaspoons soy sauce. Stir-fry on high heat, tossing frequently, until vegetables are evenly coated. Add 1 pinch salt, to taste (optional) only if needed after tasting.
Fold in eggs and green onions: Add the scrambled eggs back into the pan along with the green parts of 3 green onions (scallions), sliced, white and green parts separated. Toss gently to combine and warm through. Remove from heat and serve immediately.