Step 1: Prepare the chicken marinade. In a medium bowl, whisk together 2 tablespoons soy sauce, mirin/sake, 1 teaspoon minced garlic, grated ginger, and sesame oil. Add the chicken pieces, toss to coat, cover, and refrigerate for at least 30 minutes.
Step 2: Heat a large skillet or flat-top griddle over high heat until smoking lightly. Add 1 tablespoon neutral oil.
Step 3: Add marinated chicken to the hot pan in a single layer (cook in batches if necessary to avoid overcrowding). Sear for 2-3 minutes until a golden crust forms.
Step 4: Flip the chicken pieces and sear for another 1-2 minutes. Push chicken to the side of the pan, add 1 tablespoon unsalted butter to the empty space, let it melt, then toss chicken with melted butter and the white parts of the green onions. Cook for another 2-3 minutes until chicken is cooked through. Remove chicken and cover loosely with foil.
Step 5: Using the same pan, add 1 tablespoon neutral oil and 1 tablespoon unsalted butter. Add diced yellow onion and carrots. Sauté for 3-4 minutes until softened.
Step 6: Stir in the remaining 3 cloves minced garlic and cook for 30 seconds until fragrant. Push vegetables to the side.
Step 7: Pour beaten eggs into the empty space and scramble gently until just set. Break eggs into smaller pieces and mix with the vegetables.
Step 8: Add the chilled, day-old rice to the pan. Break up any clumps with your spatula. Drizzle with the remaining 3 tablespoons of soy sauce.
Step 9: Stir-fry constantly for 3-5 minutes, ensuring the rice is well coated, heated through, and develops some crispy edges. Season with salt and pepper to taste.
Step 10: Fold in the green parts of the green onions. Gently fold the cooked hibachi chicken back into the fried rice or serve side-by-side. Serve immediately.