Place the bones and beef brisket in a large stockpot and add enough water to cover. Bring water to a boil and let cook for 10 minutes. Drain the meat and bones in a colander and thoroughly clean the stockpot. This process removes any impurities or scum and will give you a much cleaner broth.
Place pork rib tips in one medium bowl and bok choy and mushrooms in another. Add soy sauce, chopped garlic, and chile paste to each bowl and stir to combine.
Meanwhile, working with one ingredient at a time, use tongs to char the ginger, onions, and scallions over an open flame, or place each directly on a gas burner. Turn occasionally until lightly blackened and fragrant, 2-3 minutes.
Add about 5 qts. water to the stockpot and bring to a boil. Transfer the bones and beef back to the pot, along with the charred ginger, onions, scallions, and bay leaves. Add the fish sauce and lump sugar to taste.
In a dry skillet set over medium-low heat, toast the chiles, star anise, cloves, cinnamon, cardamom, fennel, and coriander until fragrant, about 3 minutes. Use kitchen string to tie up the spices in a piece of cheesecloth and add it, along with 1 1⁄2 tsp. salt, to the broth. Cover the pot and simmer on very low heat for at least 8 hours. After 2 hours, check the brisket every hour and remove when tender. Once brisket is cool, slice into 1⁄4-inch slices against the grain and refrigerate.
Skim the broth often with a spoon to remove any foam and fat. Add lime juice and sesame oil to the broth. Taste the broth and adjust seasoning by adding more salt, sugar, and fish sauce as needed.
When the broth is finished, line a colander with cheesecloth, place over a bowl, and strain. If you want a clearer broth, strain twice. Do a final skimming of the fat.
Preheat the oven to 350°F. Place the rib tips in a roasting pan, cover with foil, and cook for 25-30 minutes. Remove foil from pan, push ribs to one side, and add bok choy and mushrooms to the pan. Turn on the broiler and broil for 10 minutes or until browned.
To serve, cook the noodles according to the package directions. Divide between serving bowls. Place a few slices of the brisket, pork rib tips, and bok choy on the noodles. Bring the broth to a rolling boil and ladle it into each bowl. The hot broth will warm the beef. Garnish with bean sprouts, fresh herbs, and be sure to squeeze a lot of fresh lime over the top!