In a small bowl, whisk together the mayonnaise, rice vinegar, tomato paste, and 1 tablespoon of soy sauce until smooth. Refrigerate until ready to serve.
Heat a large skillet or wok over medium-high heat. Add 1 tablespoon of canola oil, then the chicken. Let it cook undisturbed for 2 to 3 minutes to develop some browning, then stir and continue cooking until the chicken is cooked through, about 6 to 8 minutes total.
Reduce the heat to low. Add the butter and 1 tablespoon of soy sauce to the pan and stir to coat the chicken until glossy. Transfer to a plate and cover with foil to keep warm.
Return the heat to medium-high and add the remaining 1 tablespoon of canola oil. Add the onion, zucchini, carrot, and mushrooms. Cook, stirring occasionally, until the vegetables are tender-crisp and slightly charred in spots, about 5 to 7 minutes. Add the remaining tablespoon of soy sauce and stir to coat.
Serve the Hibachi Chicken and Vegetables over rice or noodles with the yum yum sauce on the side. Top with scallions, red pepper flakes, and toasted sesame seeds, if desired.