Steam broccoli in microwave. In a nonstick skillet, heat 1/4 inch of oil. Add the chicken and cook over moderately high heat, turning, until browned, but not cooked through. Set aside.
In a bowl, whisk the stock, oyster sauce, fish sauce, chile sauce, hoisin, sugar, and pepper flakes.
In the skillet, heat 2 tablespoons of oil. Add the bell pepper, steamed broccoli, and garlic and stir-fry over high heat until fragrant, 2 minutes. Add the noodles and stir-fry until browned, 4 minutes. Add the sauce and toss over moderately high heat, until absorbed. Fold in 1 cup of basil and the chicken. Garnish with more basil.