All praise be to Allah, peace and blessings be upon Prophet Mohammad.
If she broke fast for a lawful excuse, such as being in a state of menstruation, she is allowed to fast the six days of Shawwaal first, then make up for missed fasts since doing so is an obligation which can be offered any time before the start of next Ramadan, and the reward is attained by fasting Ramadan in addition to the six days of Shawwaal, and not because Shawwaal comes after Ramadan.
On the other hand, if she broke fast during Ramadan without a lawful excuse, it is incumbent upon her to make up for the missed days immediately after the Eid and before fasting the six days of Shawwaal. However, if she observed the fasts of Shawwaal before making up for missed fasts, then her fasting is valid but she is considered sinful, and is obliged to make up for them after that.
It is stated in [Hashiat E`anat Al-Talibeen(2/268)] what follows: “Making up for the missed days of Ramadan is an obligation that must be performed immediately if they were missed without a lawful excuse, and at convenience if the excuse was lawful.” Allah knows best.