By Horav Yosef Yeshaya Braun, member of the Badatz of Crown Heights
A non-Jew may not be a mohel (ritual circumcisor); only someone who is obligated to perform the mitzvah of bris milah (ritual circumcision) may perform it on others. If there is no Jew available to perform the bris milah, then the bris must be postponed – even beyond the eighth day – until a Jewish mohel can be found.
Someone who was circumcised by a non-Jew does not require a second bris, for there is no halachic requirement for a bris to be done lishmah (for the sake [of the mitzvah]). According to the Rema (Rabbi Moshe Isserles 16th century primary commentary on the Shulchan Aruch), however, in such a case at least hatafas dam bris ([a small amount of] blood drawn from the bris [site]) is required. There is a machlokes (difference of opinion) as to how the Mechaber (Rabbi Yosef Karo, the 16th century compiler of the Shulchan Aruch) rules. Many authorities leave room for leniency if there is merely a doubt whether the mohel was a non-Jew.
A mumar (apostate) may not perform a bris. This refers to a Jew who does not observe the entire Torah, one who is mechalel (desecrates) Shabbos, or one who is a mumar regarding the mitzvah of milah specifically – he himself not being circumcised. However, according to many opinions, even if a mumar was the mohel, hatafas dam bris is generally not required.
There are communities which ban a doctor from performing a bris, even on his own son. This was instituted as a safeguard against it becoming the standard for doctors to be mohalim, regardless of their observance of the Shabbos or their concern to perform brisim in the proper halachic manner. Strictly speaking, a Torah-observant male doctor may perform the bris—as long as he does it according to halachah—and may be the first choice if no other expert mohel is available.