Sh. al-Albānī (رَحِمَهُ اللهُ) was asked whether the prayer-caller (الْمُؤَذِّنُ) should break his fast first, or give the adhān first. The Shaykh replied that he does share such an intricately meticulous outlook on this issue. Whichever of the two the prayer-caller decides to do is permissible.

He was then asked whether the fasting person can continue to fast after the time has ended. He replied that just as it is permissible to give the adhān in the beginning of the time, it is also permissible for him to break his fast in the beginning of the time. So, whether he does it first, or afterwards, the Shaykh does not have a preference for one over the other for him to say that one should break his fast first and then give the adhān, or for him to say one should give the adhān first and then break his fast.

Source: Jāmiʿ Turāth al-ʿAllāmah al-Albānī fī al-Fiqh (10: 199).