Shall I make up the missed fasts before fasting the six days of Shawwal?

The scholars differed on this issue, and some of them said: If someone has to make up for the month of Ramadan, then he must start by making up the missed days first, in order to obtain the reward mentioned in the hadeeth, because the reward mentioned does not come until after completing the several days of Ramadan.

 The Prophet, saw, said: “Whoever observes the fast of Ramadan, and follows it with six [fasts] in Shawwal, will be [rewarded] like he fasted the entire year.” Sahih Muslim (1164)

Having said that, we can conclude that fasting the six of Shawwal before making up the missed days in Ramadan does not obtain the reward. Because the Prophet, saw, says: “Whoever observes the fast of Ramadan, and follows it with six [fasts] in Shawwal, will be [rewarded] like he fasted the entire year.”

So, it is imperative to fast the whole month, and then after that you may fast these the six of Shawwal

On the other hand, the majority of scholars believe that it is permissible to fast voluntarily before making up for Ramadan. Accordingly, it is permissible to fast the six days of Shawwal before making up missed fasts, especially for those who can make up all or most of Shawwal.

So, you actually have the leeway to finish making up missed fasts first and then fast the six days of Shawwal. Similar to a woman in Nifas who did not fast Ramadan. You can, in fact, make up the missed days and them fast the six of Shawwal, or fast the six of Shawwal first.