Wrap ginger, shallots, star anise, and cinnamon stick in two layers of cheesecloth (about 17 by 17 in.); tie with heavy cotton string. In an 8- to 10-quart pan, combine beef chuck, broth, 2 1/2 quarts water, 1/4 cup fish sauce, sugar, and spice bundle. Cover and bring to a boil over high heat; uncover, reduce heat, and simmer until beef is tender when pierced, 1 1/2 to 1 3/4 hours.
With a slotted spoon, transfer meat to a board. Remove and discard spice bundle. Skim and discard fat from broth. Add salt and more fish sauce to taste. Return broth to a simmer.
Meanwhile, arrange bean sprouts, chilies, basil, cilantro, and limes on a platter. When beef chuck is cool enough to handle, thinly slice across the grain.
Immerse sliced sirloin in simmering broth (use a wire basket or strainer, if available) and cook just until brown on the outside but still pink in the center, 30 seconds to 1 minute; lift out (with basket or a slotted spoon).
Mound hot Cooked Rice Noodles equally in deep bowls (at least 3-cup capacity). Top equally with beef chuck, sirloin, yellow onion, and green onions. Ladle broth over portions to cover generously.
Serve Beef Noodle Soup with platter of condiments, hoisin sauce, and chili paste to add to taste.