In a small saucepan, heat broth, cilantro stems, garlic, ginger, and star anise to boil, reduce to simmer. Let simmer while you cook the remaining ingredients.
Heat water and salt in a medium saucepan to boiling, add rice. Stir, reduce heat to low and cover, cook for 15 minutes.
Mix thai red curry paste with chicken, set aside.
Heat a large skillet or wok to medium high heat. Add peanut oil, heat until shimmering. Add shallot, garlic, ginger, and thai chiles to skillet, cook for 2-3 minutes or until beginning to color and soften. Add chicken, and cook until chicken is just opaque, 3-4 minutes more. Add mushrooms and bok choy, stir to ensure even cooking. Cook for 3-4 minutes or until vegetables begin to soften. In a small bowl, whisk 1 c chicken broth with cornstarch. Add chicken broth mixture, soy sauce, fish sauce, and brown sugar to skillet and stir. Turn temperature to low, and cook for 1-2 minutes or until sauce begins to thicken. Adjust seasonings to taste, adding chicken broth if sauce is too thick. Squeeze lime juice into sauce.
Serve rice with stir-fry, and sprinkle cilantro and basil on top.