Cut each chicken breast in half lengthwise. Then thinly slice the chicken breasts crosswise.
Place the sliced chicken in a large bowl and add the cornstarch, 2-3 tablespoons of soy sauce and 1 tsp of the oil.
Mix well to combine, and set aside .
In another bowl or liquid measuring cup add the water, remaining ½ cup soy sauce and brown sugar and whisk to dissolve most of the sugar.
Heat a large skillet or wok over medium/high heat.
Add 1 tablespoon oil and half of the sliced chicken.
Cook and brown the chicken on both sides until almost cooked through.
Remove to a plate and repeat with the remaining chicken, adding more oil if needed.
Return the pan to the heat with a 1 Tbs. oil and add the onion and peppers and stir fry a couple minutes.
Add garlic and ginger and stir fry an additional 30 seconds until fragrant.
Add the soy sauce mixture to the skillet, mix and bring to a boil.
Once boiling turn heat down slightly, add the chicken and simmer a few minutes until the sauce is slightly thickened and chicken is cooked through.
Stir in green onions.
Serve over hot rice and top with sesame seeds, if desired.