In a bowl or container with a lid, combine the chicken thighs and ½ cup teriyaki sauce. Cover and marinate in the refrigerator for 30 minutes to 24 hours.
Add 1 tablespoon of oil to a large (at least 12-inch) nonstick pan or wok (see notes in post about pans) and heat over medium-high heat. Carefully add the chicken and marinade (step back as it will bubble up and may spatter a bit). Cook, stirring frequently until the chicken is cooked through, about 3-4 minutes. Remove the chicken to a large bowl and wipe or clean out the pan.
Reduce the heat to medium and add another tablespoon of oil. Add the beaten eggs to the pan and cook, stirring until JUST cooked and still moist. Transfer to the bowl with the chicken and wipe or clean out the pan again.
Add a third tablespoon of oil and return the pan to medium-high heat. Add the cabbage, pineapple and red pepper, and cook, stirring occasionally, until the vegetables are crisp-tender, and the pineapple is lightly caramelized, about 2-4 minutes. Stir in the garlic and green onions and cook 30 seconds more. Transfer to the bowl with the chicken and eggs.
Wipe the pan clean and return it to the heat with one more tablespoon of oil. Add the cold rice to the pan, spread it evenly on the bottom and press it down with a spatula. Let the rice cook undisturbed for about 2 minutes (it should be sizzling).
Drizzle about 3-4 tablespoons of teriyaki sauce and 1 tablespoon sesame oil over the rice and toss to combine until the rice is heated through. Add the contents of the bowl with the chicken to the rice and stir to combine. Taste and add additional teriyaki or soy sauce as desired.
Garnish with additional green onions, sesame seeds, peanuts or cashews if desired.