Soak the brisket in a bowl of water and set aside for 10 mins to draw the blood out. Drain the water away. Add the water (4 cups), brisket, onion, green onion and black pepper into a medium pot. Cover with the lid and boil it over high heat. Skim off any scum that forms.Once the water starts rapidly boiling, reduce the heat to medium low. Simmer it until the brisket is tender and cooked. (Normally, I boil them about 1 hour, in total.) Strain the broth and cool it down for 30 mins. (This should result in about 3 cups of broth.)Take out the meat onto a plate and cool it down until safe to touch. Cut any stringy fat off the meat and thinly slice it.
Combine the beef stock (from step 1) and the dongchimi brine into a bowl / container. Add the vinegar, pear juice, sugar, salt and mustard powder to season the naengmyeon broth. Stir to mix. (Depending on your dongchimi brine and also your taste preference, you may want to adjust the quantity of these ingredients.)Cover the bowl / container and put it into the freezer and lightly freeze it (3-4 hrs). Alternatively, you can freeze it overnight and defrost in the fridge or at room temperature (if winter) until the broth is icy cold and slightly slushy when served with the noodles. If necessary, you can shave off the ice using a fork.
Boil the noodles following the package instruction. The noodles should be cooked well and the texture should be bouncy. Drain the water away and rinse the noodles under cold running water a couple of times to cool them down. If you desire, you can add some ice blocks while cooling the noodles down.Drain the water quickly and portion the noodles for serving. Place a mound of noodles into a serving bowl.
Pour the chilled naengmyeon broth over the noodles. Top it up with sliced beef, pickled radish, cucumber, pear and egg. Sprinkle some toasted sesame seeds. Serve immediately. For an extra zingy taste, you can add a little bit of mustard paste and vinegar into your soup.