Whisk the eggs in a small dish. In another small dish, stir together the soy sauce and the sesame oil; set aside. Heat 1 1/2 teaspoons of olive oil in a large skillet or wok over medium-high heat. (If you need to cook chicken to use in this dish, you may cook it at this point, remove to a plate, wipe out the pan, and then add an extra 1 1/2 teaspoons of olive oil before proceeding.) Pour in the eggs, and scramble until cooked but still a little glossy. Remove to a plate and chop.
Increase the heat to high, and heat the remaining tablespoon of oil in the same pan. Add the garlic and ginger, and sauté for 30 seconds or until fragrant. Add the cooked and cooled rice, spreading it into an even layer and pressing it down with a spatula. Sear the rice for about 2 minutes or until lightly toasted. Flip, press the mixture into an even layer again, and sear for 1-2 minutes more.
Add the mixed vegetables and green onions. Top with the chopped chicken. Before stirring, pour the soy sauce mixture over the chicken. (This allows for more even distribution of the sauce before it hits the more absorbent rice.) Then stir to fully incorporate and cook for about 2 more minutes or until all the ingredients are heated through, stirring in the eggs during the final minute. Remove from the heat and garnish with sesame seeds or peanuts, if desired.