Cook the rice according to the package directions.
In a small bowl, whisk together the peanut butter, soy sauce, lime juice, broth, sugar, and coconut milk; set aside.
Cut the chicken into 1-inch-wide strips. Heat the oil in a large skillet over medium-high heat.
Add the chicken, chili, ginger, and garlic to the skillet. Cook, stirring occasionally, until the chicken is golden, about 5 minutes.
Add the snow peas and broccoli and cook until the vegetables are just tender, about 1 minute.
Add the peanut sauce and cook until the mixture has thickened slightly and the chicken has cooked through, about 3 minutes. Serve over rice, sprinkled with cilantro.