In a small bowl, marinate the chicken pieces with 1 tablespoon soy sauce and 1 teaspoon gochugaru. Set aside for 10 minutes.
Heat 1 tablespoon vegetable oil in a large skillet or wok over medium-high heat. Add the marinated chicken and cook until golden brown and cooked through, about 5-6 minutes. Remove from the pan and set aside.
In the same pan, add the remaining oil. Add minced garlic, grated ginger, and the white parts of the green onions. Sauté for 30 seconds until fragrant.
Add the chopped kimchi and stir-fry for 2-3 minutes until it starts to caramelize.
Add the day-old rice to the pan, breaking up any clumps. Stir-fry for 3-4 minutes, allowing the rice to crisp up slightly.
Pour in the kimchi juice, remaining soy sauce, and gochugaru. Mix well to combine.
Add the cooked chicken back to the pan and stir to incorporate.
Create a well in the center of the rice mixture and crack the eggs into it. Scramble the eggs and then mix them into the rice.
Drizzle sesame oil over the rice and add the green parts of the green onions. Stir to combine.
Taste and adjust seasoning with salt if needed.
Serve hot, garnished with additional green onions if desired.