Preheat oven to 425 degrees F.
Chop green onions and white onion into slices. Mince the ginger and garlic, if not already minced.
Heat a large pan over medium/high heat. Add the sesame oil.
Once heated, add green and white onions, saute for about 3-4 minutes, until softening. Add ginger and garlic and continue to saute for about 2-3 more minutes, until onion are very soft and garlic is fragrant.
Add soy sauce, rice wine vinegar and brown sugar to the pan with the onions. Mix to combine and dissolve the sugar.
Add the chicken broth and mix to combine. Bring to a slight simmer then turn off the heat.
Add the chicken cutlets to the pan, spooning some of the sauce on top.
Place in the oven and bake for approximatley 15 minutes, until fully cooked.
Once cooked, remove chicken and place on a clean plate. Place the sauce back on the stove, over medium heat.
Bring sauce to a simmer. Combine water and cornstarch in a small bowl. Add the mixture, little by little, to the sauce, until desired thickness is reached. Be sure to immediate stir, rigorously, when adding in the cornstarch mixture.
Once sauce is thickened to your likeness, pour over the chicken.