To prepare chicken: Combine sesame oil, fish sauce, sugar, chile paste and pepper in a resealable plastic bag. Add chicken; marinate in the refrigerator for 24 hours.
To prepare broth: Warm oil over medium heat. Add garlic and ginger; cook, stirring, until fragrant and well mixed, about 2 minutes. Add water, sugar, oyster sauce and fish sauce; bring to a boil, then reduce heat and simmer 5 minutes. Pour through a fine strainer, discarding solids. Stir chile paste into broth; set aside. (You should have about 1 cup broth.).
To prepare stir-fry: Combine vegetable oil and sesame oil in a large skillet or wok; heat almost to the smoking point. Quickly add chicken; stir-fry until chicken is about half cooked. Add bell peppers, onions, mushrooms, basil and cilantro; cook until vegetables are almost tender. Add broth and bring to a boil, then add snow peas and broccoli. Cook just until crisp-tender, about 3 minutes. Dissolve cornstarch in cold water; stir into the stir-fry mixture.
Return to a boil and cook briefly until thickened.
To serve: Spoon stir-fry over rice; garnish with lime and cilantro.