Break up the chilled rice with a fork until the grains are loose. Dice the carrot and red bell pepper into ¼-inch pieces, slice the scallions (keeping whites and greens separate), mince the garlic, grate the ginger, and beat the eggs.
Heat a 12-inch nonstick skillet or wok over medium-high heat for 1 minute. Add 2 teaspoons of avocado oil, pour in the beaten eggs, and gently push them with a spatula for 45 to 60 seconds until soft curds form. Slide the eggs onto a plate.
Add 2 teaspoons of avocado oil to the same pan. Stir the carrot and red bell pepper for about 3 minutes, then add the scallion whites, garlic, and ginger for 30 seconds. Add the frozen peas and cook 1 minute, then push the vegetables to one side.
Add the final 2 teaspoons of avocado oil to the empty side of the pan, add the rice, and spread it into an even layer. Press it flat and let it sit 1 minute, then repeat the press-and-stir cycle for 4 to 5 minutes until steaming hot with lightly golden edges.
Drizzle the soy sauce around the outer edge of the pan so it sizzles, then toss for about 1 minute until the rice is evenly light tan. Return the eggs, fold them in, turn off the heat, and finish with sesame oil, white pepper, and scallion greens.
Divide among 4 bowls (about 1½ cups each), sprinkle ½ teaspoon of toasted sesame seeds over each bowl, and serve right away while hot.