There is nothing wrong, as far as Islam is concerned, in making a double intention by fasting the missed days of obligatory fasting in Ramadan as well as fasting the optional six days of Shawwal. However, some Muslim jurists state that it is recommended to make up for the missed fasts separately from fasting the six days of Shawwal so as to get extra reward. So, you are free to choose to either fast the six days of Shawwal or the missed fasting days first as the two will be in Shawwal.
Elaborating more on this issue, we’d like to cite the following fatwa issued by Sheikh `Atiyyah Saqr, former head of Al-Azhar Fatwa Committee, who states the following:
“A person who has missed days of fasting in Ramadan may fast the optional six days of Shawwal with the intention of both making up for these missed days and observing the optional fasting of six days of Shawwal. He or she will then get double benefit simultaneously: making up for the missed days and getting the reward of fasting the six days of Shawwal, for it is established in Islam that one’s acts are judged by one’s intentions.
However, it is recommended that one makes up for the missed fasts separately from fasting the six days of Shawwal [so as to get extra reward]. However, the Shafi`i scholars maintain that when one makes up for the missed fasts of Ramadan in Shawwal, one also gets the reward of fasting the optional six days of Shawwal even if one has not intended to fast those six days principally; yet the reward of fasting the six days here will be less than if one has intended to fast them from the beginning.
According to Ash-Sharqawi `Ala At-Tahrir by Sheikh Zakariyah Al-Ansari, (vol. 1, p. 427) when a Muslim makes up for missed fasts of Ramadan in Shawwal or fasts some days he or she has vowed to observe in Shawwal, or even offers optional fasting in Shawwal other than fasting the six days recommended to be observed in Shawwal, he or she will also get the reward of fasting the six days of Shawwal. This is because the point is to fast any six days of Shawwal following the fast of Ramadan. But one then will not get the whole reward of principally intending to fast the six days of Shawwal specifically. It is to be noted that this does not apply to the person who has missed fasting the whole month of Ramadan and made up for it in Shawwal for he or she then does not fall under the category the Prophet (peace and blessings be upon him) referred to in the hadith: “Whoever fasts the month of Ramadan, and then follows it by (fasting) six days of Shawwal, it is as though he has fasted the whole year.”
This is parallel to the juristic point of view regarding the recommendation of greeting the mosque when entering it by offering two rak`ahs before sitting. This recommendation is accomplished any time one enters a mosque and prays two rak`ahs, whether one intends merely to perform the prescribed prayer or any two supererogatory rak`ahs, for the point here is to offer a prayer before sitting in the mosque, and this is fulfilled by observing the prescribed prayer or the supererogatory one.
According to the author of Al-Bahgah, one gets the reward of greeting the mosque by offering any prayer before sitting, even if one has not intended that this prayer be for greeting the mosque.
But it is to be borne in mind that one in this case does not deny having the intention of greeting the mosque.
Based on the above, it is permissible for one to fast six days of Shawwal intending simultaneously both to make up for the fasts one has missed in Ramadan and to offer the recommended fasting of six days in Shawwal, especially if one is interested in getting the reward of fasting the six days of Shawwal with making up for the missed fasts of Ramadan but finds it exhausting to do both separately.
Besides, if one in this case intends only to make up for the missed Ramadan fasts in Shawwal (and it happens that these days are six or more), one will get also the reward of fasting the six days of Shawwal.
The supererogatory act of fasting the six days of Shawwal here is sub-categorized under the obligation of making up for the missed fasts of Ramadan. This is an alleviation reasoned by jurists, and hence, there is no need for adopting a view of a certain school in this regard and judging the other views as wrong.
The wisdom behind recommending fasting six days of Shawwal following a whole month of fasting in Ramadan is that Muslims gradually shift from a state of abstaining from food or other desires for a long hours on successive days to a state of eating food and satisfying the other worldly lawful desires whenever one wants. A sudden shift in this case may harm a person’s health.”