Heat 1 tbsp oil in a large skillet or wok over medium-high heat. Add onion and carrot; cook until soft, about 3–4 minutes.
Push veg to the side, add remaining oil, and pour in beaten eggs if using; scramble quickly and mix with veg.
Add garlic, then the chilled rice. Break up clumps and stir-fry until heated through, about 3–5 minutes.
Toss in chopped greens and soy sauce; stir until greens wilt and everything is evenly coated. Finish with sesame oil, salt, and pepper.
Taste and adjust seasoning. Serve immediately with scallions.