In a small bowl, stir together soy sauce, sesame oil, rice wine vinegar, and sugar (if using). Set aside.
Heat a large skillet or wok over medium-high heat. Add 1 tablespoon of oil and heat for 30 seconds until shimmering. Add the onion, mushrooms, carrots, and celery. Sauté until the vegetables start to soften. About 3-5 minutes.
Add the garlic and cook for another 30-60 seconds, or until fragrant. Using a slotted spoon, transfer the veggies to a bowl and keep warm. If anything stuck to the bottom, add a little water to the pan to help release it and pour the liquid into the bowl with the veggies.
Add a little more of the oil to the pan or wok. Add the lightly beaten eggs and scramble quickly using a spatula or chopsticks. Place cooked eggs in a bowl and set aside.
If egg has stuck to the bottom of the pan or wok, give it a quick rinse or wipe it out. Heat the wok again and add 1 tablespoon oil. Add the broccoli and saute until bright green in color, but still crisp, about 3-5 minutes. Transfer to bowl with other veggies.
Add the remaining oil to the pan or wok, heat until shimmering, then add the cold rice to the pan. Cook, stirring or turning rice over constantly, until it is lightly toasted and hot throughout, about 5-10 minutes. Add the vegetables and eggs to the pan. Stir well.
Drizzle the soy sauce mixture evenly over the fried rice and stir to make sure everything is coated with the sauce. Taste and add extra soy sauce or sesame oil, if needed.
To serve, spoon the fried rice into bowls and sprinkle with green onions. Eat while warm.