1 small onion, finely chopped
8 garlic cloves, finely grated
1 (2-inch) piece ginger, peeled, finely grated
1/4 cup gochujang (Korean hot pepper paste)
1/4 cup soy sauce
2 tablespoons sugar
1 tablespoon mirin
1 tablespoon toasted sesame oil
1 teaspoon freshly ground black pepper, plus more
3 cups cooked short-grain rice
1 teaspoon ground cumin
1/2 teaspoon garlic powder
1/2 teaspoon mustard powder
8 chicken drumsticks, patted dry
4 tablespoons unsalted butter
2 tablespoons vegetable oil
4 cups low-sodium chicken broth
6 scallions, white and pale-green parts only, cut into 1-inch pieces
Mix onion, garlic, ginger, gochujang, soy sauce, sugar, mirin, sesame oil, and 1 tsp. pepper in a medium bowl to combine; set sauce aside.
Toss rice, cumin, garlic powder, and mustard powder in a medium bowl to combine; spread out on a parchment-lined rimmed baking sheet and chill until cold, about 1 hour.
Meanwhile, season chicken generously with salt. Heat butter and vegetable oil in a medium heavy pot over medium. As soon as foaming subsides, add chicken and cook, turning occasionally, until lightly browned, about 5 minutes. Add reserved sauce and bring to a simmer; cook until sauce appears to thin out, about 3 minutes. Add broth and bring to a boil. Reduce heat, partially cover pot, and simmer, turning occasionally, until chicken is very tender, 45–55 minutes.
Divide rice into 4 portions; form into 3/4"-thick disks (moisten hands with water to prevent sticking). Heat a large nonstick skillet over medium. Working in 2 batches, cook rice cakes, turning halfway through, until crisp, puffed, and golden, about 10 minutes. Transfer to plates.
Add scallions to chicken and cook until tender, about 3 minutes. Season with salt and pepper. Top each cake with 2 chicken legs and a few scallions; spoon plenty of sauce over.