Trim and slice the chicken into thin fillets. Season on both sides with the spice mix.
After the rice has been cooking for 10 minutes, heat the butter and olive oil in a large skillet.
Add the chicken, in batches if necessary, and cook for a couple minutes on both sides. It won't take long to cook through because the pieces are thin.
Remove the chicken to a plate while you make the pan sauce.
Add the lime juice and chicken broth to the skillet. Use a whisk to mix up any browned bits on the bottom of the pan.
Simmer gently and allow to thicken slightly. Add more butter if you want the sauce to be richer.
Keep an eye on the rice and remove from the heat if needed.
Return the chicken and any juices to the pan.
Turn the chicken in the pan sauce to coat well.
Before serving, stir in a generous handful of chopped cilantro to the rice.