In a medium bowl, whisk together the soy sauce, sake (or mirin), sesame oil, minced garlic, grated ginger, sugar, and black pepper.
Add the cubed chicken to the marinade, ensuring each piece is well coated.
Cover the bowl and refrigerate for at least 30 minutes, or up to 4 hours for a deeper flavor.
Heat 1 tablespoon of sesame oil in a large skillet or wok over medium-high heat.
Pour in the beaten eggs and cook, stirring constantly, until scrambled. Remove the eggs from the pan and set aside.
Add another tablespoon of sesame oil to the pan. Add the chopped onion and cook until softened, about 2-3 minutes.
Add the mixed vegetables and cook for another 3-4 minutes, until slightly tender.
Add the cooked rice to the pan and break it up with a spatula. Stir-fry for 2-3 minutes.
Pour in the soy sauce and add the scrambled eggs back to the pan. Stir everything together until well combined and heated through.
Preheat your grill or skillet over medium-high heat.
Remove the chicken from the marinade, letting any excess drip off.
Grill or sauté the chicken in batches, being careful not to overcrowd the pan. Cook for 5-7 minutes, or until the chicken is cooked through and lightly browned on all sides. Set aside.
Add the zucchini, carrots, broccoli, onion, and bell pepper to the grill or skillet. Cook, stirring occasionally, until the vegetables are tender-crisp, about 6-8 minutes.
In a small bowl, whisk together mayonnaise, water, tomato paste, garlic powder, paprika, cayenne pepper (optional), melted butter and sugar until smooth. Refrigerate for at least 30 minutes to allow the flavors to meld.
Divide the fried rice among plates. Arrange the grilled chicken and vegetables alongside the rice. Drizzle generously with yum yum sauce. Garnish with sesame seeds and chopped green onions, if desired.
