Cook chicken in a 6-quart pot of boiling unsalted water, covered, until just cooked through, about 3 minutes. Transfer to a large bowl with a slotted spoon.
Add broccoli to boiling water and cook, uncovered, stirring occasionally, until just tender, 3 to 5 minutes. Transfer with slotted spoon to a colander to drain, then transfer to another bowl. Return water to a boil and cook noodles until tender (check often; cooking time on package may not be accurate). Reserve 1 cup cooking water, then drain noodles in colander and rinse under hot water.
While noodles cook, tear chicken into chunks.
Add oyster and hoisin sauces, sesame oil, chile garlic paste, half of scallions, and 1/3 cup cooking water to chicken and stir to combine.
Divide noodles, broccoli, and chicken mixture among 4 bowls and sprinkle with sesame seeds and remaining scallions. Serve immediately, stirring just before eating. If noodles become dry, moisten with some of cooking water.