1. Cook chicken in a 6 quart pot of boiling unsalted water, covered, until cooked through, about 3 minutes. Transfer to a large bowl with a slotted spoon.
2. Add broccoli to boiling water and cook, uncovered, stirring occasionally, until just tender, 3-5 minutes. Transfer with a slotted spoon to a colander to drain, then transfer to another bowl.
3. Return water to boil and cook noodles until tender (check often; cooking time on package may not be accurate).
4. Reserve 1/2 cup cooking water.
5. Drain noodles in colander and rinse under hot water.
6. Tear chicken into chucks.
7. Add oyster and hoisin sauces, sesame oil, chile garlic paste, half of scallions and 2 tablespoons of cooking water to chicken and stir to combine.
8. Divide noodles, broccoli and chicken mixture among 2 bowls.
9. Sprinkle with sesame seeds and remaining scallions. Serve immediately, stirring just before eating. If noodles become dry, moisten with some of the cooking water.