Pat the chicken wings dry with paper towels and cut each into drumette and wingette (save the wing tips for making broth).
Heat the cooking oil in a wok and saute the garlic and ginger until aromatic.
Add the mushroom caps and carrot, and continue sauteeing for a minute.
Dump the chicken into the wok and cook, stirring often, until the meat is no longer pink.
Pour in the soy sauce, rice wine and oyster sauce. Stir in the sugar.
Measure a cup of the mushroom soaking water and add to the chicken.
Bring to the boil, lower the heat, cover the wok and braise the chicken and shiitake for about 20 minutes.
Midway through the cooking, taste the sauce. If it needs more saltiness, add salt (see notes after the recipe). If more sugar is needed for balance, add a pinch or two of sugar.
Disperse the starch in two tablespoons of water and stir in the sesame seed oil.
Pour the starch solution into wok and stir until the sauce is thickened and no longer cloudy.
Scoop the braised chicken wings and shiitake onto a bowl and serve immediately with hot rice.