Combine all the ingredients for the sauce in a bowl. Stir until smooth and set aside. For the best flavor, refrigerate the sauce for at least an hour before serving.
Heat a wok or large skillet over medium-high heat. Add 1 tablespoon of butter and 1 tablespoon of sesame oil, allowing the butter to melt and bubble.
Add the chicken pieces to the hot skillet. Season with salt and pepper, cooking and turning occasionally until the chicken is browned on all sides and fully cooked, about 5-8 minutes.
Reduce the heat to medium. Add the minced garlic and cook for about 30 seconds until fragrant. Then, pour in the soy sauce, brown sugar, and the remaining butter. Stir to coat the chicken evenly.
Squeeze the juice of half a lemon over the chicken and stir once more. Let the mixture simmer for 2-3 minutes until the sauce thickens and coats the chicken.
In a separate skillet or the same wok after cooking the chicken, heat 1 tablespoon of sesame oil over medium-high heat. Add the zucchini, mushrooms, onion, and carrots. Season with salt and pepper. Stir-fry for about 5-7 minutes until the vegetables are tender yet still crisp. Add 1 tablespoon of soy sauce and stir to coat the vegetables evenly. Cook for another 1-2 minutes.
In the same skillet or wok, push the vegetables to one side and add the beaten eggs. Scramble until fully cooked, then mix them with the vegetables. Add the cooked rice, thawed peas, and carrots to the skillet, stirring well to combine.
Add the soy sauce, sesame oil, and butter. Stir to coat the rice evenly. Cook for another 5-7 minutes, stirring occasionally, until the rice is heated through. Add the minced garlic and chopped green onions. Stir and cook for an additional 2-3 minutes.
Serve the hibachi chicken with the cooked vegetables and fried rice. For extra flavor, drizzle with the remaining sesame oil. Pair the dish with the Yum Yum Sauce for dipping.