Combine water and rice in a saucepan over medium-high heat; bring to a boil. Reduce heat to low, cover, and simmer until rice is tender and liquid is absorbed, about 20 minutes.

Meanwhile, stir together soy sauce, peanut butter, vinegar, and cayenne pepper in a small bowl; set aside.

Heat olive oil in a skillet or wok over high heat. Cook and stir chicken, garlic, and ginger in hot oil until chicken is lightly browned, about 5 minutes. Reduce heat to medium and add broccoli, green onions, peanuts, and soy sauce mixture. Cook, stirring frequently, until broccoli is tender and chicken is cooked through, about 5 minutes.

Serve chicken-broccoli mixture over cooked rice.