1. Bring broth to a simmer in a 3- to 4-quart saucepan. Add chicken; simmer gently until cooked through, about 20 minutes.

2. Meanwhile, bring a large pot of water to a boil. Salt generously, add egg noodles, and cook until al dente, 5 to 7 minutes. Drain and divide noodles among 4 soup bowls.

3. Remove chicken from broth and set aside to cool slightly. Bring broth to a boil, add broccoli, and cook until crisp-tender, 3 to 4 minutes. Using a skimmer or slotted spoon, lift out broccoli and divide among soup bowls.

4. Slice chicken and divide among bowls. Ladle broth into bowls, and top each serving with a poached egg and some hot sauce, if desired.