If rice cakes are frozen, soak them in water for about 20 minutes in a medium-sized bowl. This should thaw them out a bit.
Heat oil in a large nonstick skillet over medium high heat. Add beef. With the back of a wooden spoon, break up the beef as it cooks. Once the beef starts to brown, add onions, garlic, ginger, salt, and green onions. Cook until the beef is brown and the onions start getting soft.
Add kochujang and stir until the beef and onion mixture is coated. Take the skillet off the heat and set aside.
Drain the rice cakes. In a medium-sized saucepan, add the rice cakes and add enough water to just cover the rice cakes. Bring to a boil and let cook for about 1 minute.
With a large slotted spoon, scoop the rice cakes into the skillet with the beef mixture. Add the fish cakes. Toss gently until everything is coated. Feel free to add about 1/2 cup water (or chicken broth) to make the mixture a little more saucy.
Pour the beef and rice cake mixture into a medium-sized casserole dish (2 quart size). Stir in 1/2 cup of mozzarella.
Top the dish with the remaining mozzarella and Colby cheese.
With the oven broiler on high, place the dish on the middle rack of the oven. Broil until the cheese is golden and bubbly or about 6-8 minutes.