Is the muezzin to break his fast and then make the call to prayer, or make the call to prayer and then break his fast, given that people rely on him for the Maghrib call to prayer at the very first moment of sunset?

The criterion for a fasting person to break their fast is the certainty or strong presumption that the sun has set. It is permissible to follow a trustworthy muezzin who is knowledgeable of the prayer times. Hastening to break the fast after verifying sunset is a Sunnah. It is more appropriate for the muezzin to break their fast before calling the Adhan, and not to delay breaking their fast for the sake of others, because prioritizing others in acts of worship is disliked by some scholars. The muezzin can break their fast with something small and then call the Adhan.