Heat 1 tablespoon of vegetable oil in a large wok or skillet over medium-high heat. Add the diced chicken and cook until browned and cooked through, about 5 to 7 minutes. Remove the chicken from the pan and set aside.
In the same skillet, pour in the beaten eggs. Scramble them quickly until fully cooked, then remove them from the pan and set aside with the chicken.
Add the remaining 1 tablespoon of oil to the skillet. Toss in the diced red bell pepper, green peas, and the white parts of the green onions. Stir-fry for 2 to 3 minutes until the vegetables are tender.
Increase the heat to high. Add the butter and let it melt, then add the chilled day-old rice. Break up any clumps and spread the rice evenly across the pan. Let it cook undisturbed for about 1 minute to develop crispy edges, then stir and repeat.
Return the cooked chicken and scrambled eggs to the skillet. Pour the soy sauce and toasted sesame oil over the rice. Stir everything together until well combined and heated through.
Garnish with the green parts of the green onions. Serve hot with fresh cucumber slices on the side.