Place beef brisket in a large pot and fill half of the pot with water. Cover and bring to a rapid boil over high heat. Cook for 5 minutes, removing impurities from beef. Remove from heat, drain and rinse with cold water. Wash the pot. Return brisket to the clean pot with 1 gallon of water. Cover and bring to a boil over high heat. Lower heat to medium-low and simmer for 1 hour.
Add onion to broth. Cook for 2 more hours, adding more water as it evaporates.
Meanwhile, prepare the sauce by combining gochugaru with chili oil, sesame oil, garlic, and soy sauce. Set aside.
After 2 hours, turn off the heat and remove onion and beef from soup, discarding onion. There should be about 10-11 cups of broth remaining. Skim the fat.
Shred beef brisket using two forks. The meat should be tender enough to easily fall apart. Mix shredded beef, bean sprouts, royal fern, and white stem of the green onions with prepared sauce in a large bowl. Mix to combine.
Bring soup back to a boil. Add beef mixture, lower heat to medium-low, and simmer for 30 minutes.
Bring a medium pot of water to a boil. Add sliced leeks and blanch for 30 seconds. Drain immediately soak in ice cold water. Squeeze out water and set aside.
Add top green parts of the green onions and leeks to the soup. Simmer for an additional 10 minutes.
Cook sweet potato noodles according to directions on package.
Slowly add whisked eggs into the soup, stirring in clockwise motion. Cook for another minute. Season soup with salt and pepper. Portion the noodles into 6 bowls, ladle the soup and serve.