In a large straight-sided skillet over medium-high heat, heat oil. Add bell pepper, shallot, and garlic and cook, stirring, until fragrant, about 2 minutes. Increase heat to high and add chicken, chile, and white pepper. Cook, breaking up meat with a wooden spoon and stirring occasionally, until chicken is just cooked through and any liquid is absorbed, 3 to 5 minutes.
Add green beans, oyster sauce, fish sauce, and brown sugar and stir to combine. Cook, stirring frequently, until green beans are crisp-tender, about 1 minute. Remove from heat and stir in basil until just wilted, about 20 seconds. Scrape chicken mixture into a large bowl.
Return skillet to medium-high heat and cook rice, stirring constantly, until very lightly toasted, about 30 seconds. Pour in milk, water, lime juice, and ½ teaspoon salt. Bring to a boil, cover, and reduce heat to low. Gently simmer until rice is tender and liquid is absorbed, 16 to 18 minutes. Fluff rice with a fork and return chicken mixture to skillet, spooning over rice. Cover and cook until chicken mixture is heated through, about 1 minute more.
