Bris milah or Bircas Hachama which comes first?

R' Elyashiv was asked this question by one of his grandchildren who had a baby last Wednesday. R' Elyashiv's psak was that first you do Bircas Hachama and then the mila. The reason for the pask is not given but I would guess that it has to with the fact that Bircas Hachama is a mitzva that has a very short time (only the first 3 hours) while milah is all day.