Whisk all the Sauce ingredients in a 2-cup measuring cup or small bowl and set aside.
In a wide, shallow skillet, heat 1 tablespoon of the oil over medium-high heat.
Once the oil is hot, add half of the chicken. Cook until browned and just cooked through. Remove onto a plate or bowl. (Adding all the chicken at once would steam the chicken and no browning would occur.)
Add the carrots and cook until almost tender, about 5 to 7 minutes.
Add the remaining chicken to the carrots; cook until the chicken is browned and just cooked through, adding more oil as necessary.
Add water chestnuts and the previously cooked chicken, and any of its juices, back to the pan.
Stir the sauce to recombine and add to the pan.
Stir well to coat all the ingredients and allow the sauce to come up to a boil to thicken.
Be sure your chicken is fully cooked at this point. Taste for seasoning and adjust as desired.
Sprinkle with green onions, sesame seeds, and peanuts if desired.
Serve with hot Jasmine rice.