Soak the rice cakes in a big bowl of cold water. Use enough water to cover them. Soak for at least 30 minutes to one hour if possible. One hour is best. You are doing this to help the rice cake soften up. (Even if you buy fresh rice cake, my mom says to soak them for a while so they don’t stick together when you cook them. If soaking overnight, use cold water and put in the fridge).
In a large wok, heat the sesame oil over medium high heat. Add the pork, season with salt and pepper and cook until browned, about 5 minutes. Add the water, brown sugar, gochujang, carrots (if using) and duk (rice cake). You want enough water/liquid to barely cover all the rice cake. Bring to a boil and then reduce to low and simmer for about 25-30 minutes.
After the sauce has thickened slightly, add the green onions and cook for another 5 minutes or so. Taste your sauce! Add more sugar and/or gochujang if needed. If the sauce is too thin, mix the flour or cornstarch with a little water in a small bowl and then add the mixture to the sauce and stir in well. At the same time, boil water for your eggs and boil the eggs for about 6 minutes.
To serve: garnish with more green onion, sprinkle sesame seeds and add your hard-boiled egg (peeled and cut in half) on the side. Serve immediately.