Combine first low sodium soy sauce (2 tablespoons) with vegetable oil, garlic, ginger and green onion. Add pork to marinade. Let sit 30 minutes.
Meanwhile, blanch broccoli. Make sauce by whisking together broth, cornstarch, additional 2 tablespoons low sodium soy sauce, hoisin sauce and sherry. Set aside.
Drain marinade from pork. Discard the marinade.
Heat 1 teaspoon of vegetable oil in a skillet. Add pork and cook until opaque, about 1 minute. Transfer to a plate.
Heat another teaspoon of vegetable oil in a skillet and add broccoli, peppers and water chestnuts. Stir fry until tender crisp. Add pork and sauce. Continue to cook until sauce thickens, about 3 minutes.
Serve 1 cup stir fry over 1/2 cup rice. Garnish with additional green onion if desired.