Step 1

Mix onion, garlic, ginger, gochujang, soy sauce, sugar, mirin, sesame oil, and 1 tsp. pepper in a medium bowl to combine; set sauce aside.

Step 2

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.

Step 3

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.

Step 4

Divide rice into 4 portions; form into ¾"-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.

Step 5

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.