Bring rice, water, and salt to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and liquid has been absorbed, 20 to 25 minutes.
Heat 2 tablespoons sunflower oil in a wok or skillet over medium-high heat; cook and stir chicken, garlic, and red chile pepper until chicken is golden brown, 5 to 7 minutes. Stir in curry paste until fragrant, about 1 minute. Add cooked rice and green onions, cooking and stirring for 5 minutes more. Season with soy sauce.
Push rice mixture to one side of the wok. Heat remaining 1 teaspoon sunflower oil in middle of wok; cook and stir eggs until just set, about 1 minute. Stir rice mixture into eggs. Sprinkle with peanuts and cilantro. Serve immediately.