Whisk the sesame oil, soy sauce, hoisin sauce, minced garlic, and grated ginger together in a small bowl until smooth. Pour the mixture over the cubed chicken and toss so every piece is coated.

Warm a large skillet over medium high heat and add the chicken with all of its marinade. Cook for roughly 10 minutes, stirring now and then, until the pieces are cooked through and lightly browned. Move the chicken to a plate and wipe the skillet clean with a paper towel.

Prepare the rice following the package directions on the stovetop or in a rice cooker, or use rice you cooked up to 2 days ahead and chilled. While the rice cooks, stir the mayonnaise, soy sauce, rice vinegar, and Dijon mustard together for the dipping sauce and set it aside.

Return the skillet to the heat and add the olive oil. Toss in the sliced carrot, mushrooms, onion, and zucchini, then season with salt, pepper, and the soy sauce. Cook for about 5 minutes, just until the vegetables are tender but still have a little bite. Slide them onto a plate and set aside.

In the same skillet, melt the butter and saute the garlic for about 30 seconds, just until it smells fragrant. Add the cooked rice and soy sauce and stir constantly for 5 minutes so the rice heats through and soaks up the flavor.

Push the rice to the edges to open a well in the middle and crack the egg into it. Scramble the egg in that space until it is barely set, then fold it into the rice. Stir in the sliced green onions and pull the skillet off the heat.

Divide the fried rice among plates and top with the chicken and sauteed vegetables. Serve the mustard sauce on the side for dipping or drizzling.