Heat a wok or a large saucepan over high heat until very hot. Add 1 tablespoon of the oil, and then add the broccoli, carrots, cabbage, zucchini, peppers and edamame. Stir often until the vegetables are crisp-tender, about 4 minutes. Remove them to a plate and set aside.
Heat 1 more tablespoon of oil in the same wok or saucepan over high heat. Add the beaten eggs and scramble quickly, just about 1 minute in all, sliding them out onto the plate with the vegetables when they are just cooked through.
Return the wok to the burner over high heat, add the remaining tablespoon of oil, then add the cooled rice and stir fry, stirring occasionally, until it is lightly browned in spots, about 5 minutes. Add ginger and the garlic and stir until you can smell the seasonings, about 1 minute. Add the soy sauce, sesame oil and 1/2 cup scallions, then dump in the cooked vegetables and eggs and cook until everything is well combined and hot and the vegetables are just tender, about 3 minutes, stirring often.
Garnish with the additional scallions and serve right away, passing soy sauce on the side for those who want a bit more.