Heat the avocado oil over medium-high heat in a large skillet or wok.
Add the diced chicken, salt, and pepper and arrange in a single layer. Let cook, undisturbed, until a golden brown crust forms, 3 to 4 minutes. Toss and continue to cook until the chicken is cooked through, 1 to 2 more minutes. Use a slotted spoon to transfer the chicken to a clean plate. Set aside.
Add the butter and melt, then add the carrots, onions, green onions, garlic, and ginger. Cook, stirring, until the veggies are tender, about 4 minutes.
Add the day-old rice and toss until well combined with the veggies. Spread the rice into a single layer, press down with your spatula, and cook until golden brown on the bottom, about 3 minutes.
Add the tamari and toasted sesame oil and cook, stirring to combine, and allow the sauce to soak into the rice, about 2 minutes.
Add the chicken back into the pan and toss to combine. Push the rice to one half of the skillet and pour the eggs on the empty half. Cook the eggs, stirring, until softly scrambled, 1 to 2 minutes. Stir the scrambled eggs into the rice until well combined.
Remove from the heat and stir in the toasted sesame seeds. Taste and add more salt, if desired.