Add the onion, garlic, ginger, gochujang, soy sauce, sugar, mirin, sesame oil and 1 tsp pepper to a small blender and blend until smooth.
Cook the rice, drain and then toss in a bowl with the cumin, garlic powder and mustard powder:; spread out on a parchment-lined rimmed baking sheet and chill until cold, about 1 hour.
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 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.
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.