Heat a large cast-iron skillet over high. Whisk together soy sauce, broth, lime juice, and cornstarch. Season pork with salt, pepper, and brown sugar.

Swirl 2 tablespoons oil into skillet. Add pork and cook, stirring occasionally, until browned in places and just cooked through, 2 to 3 minutes. Transfer to a plate. Swirl remaining 2 tablespoons oil into skillet. Add broccoli, onion, and garlic; reduce heat to medium-high. Cook, stirring frequently, until vegetables are browned in places, about 5 minutes. Add ginger and chili paste; cook 30 seconds. Add soy mixture; simmer until sauce is reduced and thickens slightly and broccoli is crisp-tender, about 2 minutes.

Return pork and accumulated juices to skillet along with water chestnuts, stirring until combined and sauce evenly coats pork and vegetables, about 1 minute. Serve over rice.