Step 1

Slice one of the onion quarters as thin as possible; set aside. In a large pot, bring the remaining 3 onion quarters, beef broth, chicken broth, water, fish sauce, ginger, cinnamon stick, sugar, star anise, and cloves to a boil. Reduce the heat to medium-low and simmer, uncovered, for 30 minutes.

Step 2

Meanwhile, bring a large pot of water to a boil. Off the heat, add the noodles. Let sit, stirring regularly to prevent sticking, until the noodles are soft and pliable but still not tender, 5 to 10 minutes. Drain and rinse well with cold water. Set aside.

Step 3

Over a large bowl, strain the broth through a fine mesh strainer. Discard the solids and pour the clear broth back into the pot. (Alternatively, you can use a fine mesh spider to fish out the solids.)

Step 4

Bring the broth to a boil and add the noodles; cook until the noodles are just tender, 30 seconds to 1 minute. Using tongs or a spider, lift the noodles from the pot and divide evenly into bowls. Reduce the heat to a simmer and add the sliced beef to the broth; let simmer until no longer pink, about 1 minute. Using a slotted spoon or spider, remove the beef from the broth and divide between the bowls. Taste the broth and adjust the seasoning with salt, if necessary. Ladle the broth over the noodles and beef. Top each bowl with the reserved thinly sliced onions, bean sprouts, and cilantro (you may not need all of the onions). Pass the lime wedges, hoisin, and sriracha at the table.