Combine gochujang, gochu-garu, soy sauce, sugar, milk, heavy cream, and the water in a bowl. Mix well until it has no lumps and becomes orange color. Set it aside.
Drain the rice cakes. Heat up a thick bottomed pan over medium heat. Add bacon and stir for about 5 minutes until crispy and it turns golden brown. Turn off the pan and pour out the excess fat from the bacon into a small ball. Discard the fat. Turn the heat on to medium-high. Add the garlic and onion to the bacon and stir for 1 minute. Add the sausage and the rice cake, stirring with a wooden spoon for a minute, until the sausage splits a little along the slits. Add the rice cake and 1 cup water and stir. Cover and cook for 5 minutes until the rice cake gets tender. Open and add the sauce, mixing well with the wooden spoon. Cook another 5 minutes, stirring occasionally. Add mozzarella cheese and stir to melt.
Stir and ladle some of the tteokbokki, egg, and broth into bowls. Sprinkle some chopped green onion over top and serve right away.
