Should the muezzin break his fast first or give the adhaan?

The basic principle is that the person who is fasting should break his fast after the sun has set and night has begun, because Allaah says (interpretation of the meaning): 

“and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night), then complete your Sawm (fast) till the nightfall”

[al-Baqarah 2:187]

al-Tabari said: 

With regard to the words “then complete your Sawm (fast) till the nightfall”, Allaah defines the end of the fast as being when night comes, and He defines the time for breaking the fast, when eating, drinking and intercourse become permissible. And the fast begins when the day starts and the night disappears. This indicates that there is no fasting at night, and there is no breaking the fast by day during the days of fasting. End quote. 

Tafseer al-Tabari, 3/532. 

The Sunnah is to hasten to break the fast for those who are fasting. 

It was narrated from Sahl ibn Sa’d (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “The people will continue to do well so long as they hasten to break the fast.” Narrated by al-Bukhaari, 1856; Muslim, 1098. 

Ibn ‘Abd al-Barr (may Allaah have mercy on him) said: 

It is Sunnah to hasten to break the fast and to delay sahoor. Hastening should only be done after ascertaining that the sun has indeed set. It is not permissible for anyone to break the fast when he is not sure whether the sun has set or not, because if an obligatory act becomes binding when there is certainty, then he cannot stop doing this deed except when there is certainty. End quote. 

Al-Tamheed, 21/97, 98. 

Al-Nawawi (may Allaah have mercy on him) said: 

This shows that we are encouraged to hasten to break the fast after ascertaining that the sun has set. What this means is that the ummah will continue to be in good shape and they will be fine so long as they adhere to this Sunnah. End quote. 

Sharh Muslim, 7/208. 

With regard to the muezzin, if there are people who are waiting for his adhaan in order to break their fast, then he should hasten to give the adhaan so that he will not be a cause of the people delaying to break their fast, because that is contrary to the Sunnah. 

But if he breaks his fast with something simple (such as drinking water), that will not cause him to delay the adhaan, then there is nothing wrong with that. 

If no one is waiting to hear the adhaan of the muezzin, such as if he is giving the adhaan for himself (such as a man who is alone in the desert), or he is giving the adhaan for a group of people who are close at hand (such as a group of travellers), then there is nothing wrong with him breaking his fast before giving the adhaan, because his companions will break the fast with him even if he does not give the adhaan, so they are not waiting for his adhaan. 

And Allaah knows best.