Make the marinade: Finely chop ¼ medium white onion. Peel one 1" piece ginger with a spoon. Smash 3 garlic cloves. Slice ginger and garlic. Transfer to a blender and add 5 Tbsp. gochujang (Korean hot pepper paste), 3 Tbsp. unseasoned rice wine or dry sherry, 2 Tbsp. raw or light brown sugar, 2 Tbsp. soy sauce, 1 Tbsp. gochugaru, and 1 tsp. mild Korean curry powder. Purée on high until a loose paste forms (this may take up to 2 minutes depending on your blender). Transfer marinade to a large bowl.
Cut 1 lb. skinless, boneless chicken thighs and cut into 1" cubes, add to marinade, and toss to coat. Cover bowl and chill at least 30 minutes and up to 36 hours.
Meanwhile, slice ¼ small green cabbage as thinly as possible. Peel 1 small sweet potato, halve lengthwise, then slice crosswise into 1/8"-thick half-moons. Trim stems from 8 Thai purple basil leaves, then cut leaves into quarters.
If using fresh rice cakes, place 36 rice cakes in a large bowl and pour in water to cover. Soak until rice cakes are soft, 7–10 minutes. Drain well in a colander. If using frozen rice cakes, bring a large saucepan or small pot of water to a boil and drop rice cakes into pot, stirring constantly to make sure they don’t stick together. Cook 1 minute, then drain well in a colander. Drizzle ½ tsp. extra-virgin olive oil or vegetable oil over rice cakes and toss to coat (this will prevent them from sticking together).
Heat remaining 3 Tbsp. extra-virgin olive oil or vegetable oil a large skillet over medium. Combine chicken and any remaining marinade, cabbage, sweet potato, and rice cakes in skillet and cook, stirring often, until chicken is cooked through and sweet potato is tender, 15–17 minutes. Add quartered basil leaves and cook, stirring, until slightly wilted, about 30 seconds.