Commentary for Leviticus 5:8

והקריב את אשר לחטאת ראשונה AND HE SHALL OFFER THAT WHICH IS FOR A SIN-OFFERING FIRST — This verse establishes the general rule that the sin-offering is always sacrificed before the burnt-offering. — To what may this be compared? To the case of an advocate (in our case the חטאת) who went in to the king to obtain pardon for his client. When the advocate has gained that pardon, then the present (עולה) is brought in after him (Zevachim 7b).

ולא יבדיל; he only needs to perform the procedure on either the windpipe or the foodpipe.

One organ. In the first chapter of Chulin (21b) Rashi explains because it is written: “He does not separate,” and a bird is slaughtered with only one siman (gullet or windpipe). If he does anything more than what is necessary it is considered “separation.” A further consideration: the [severing of the] entire organ is included in the mitzvah of slaughtering a bird in the preferred way — therefore, regarding shearing [the head], when he nips the entire siman it is not “separation.” With regard to the second siman, however, there is not even a mitzvah [of slaughtering] in the preferred way, as [the Rabbis] taught [in a Baraisa] (ibid. 27b): To obligate him in two simanim is impossible, for a bird was already compared to fish.

והקריב את אשר לחטאת ראשונה, “he (the priest) is to present the offering serving as the sin offering, first;” the verse does not discuss what is offered first on the altar, but what is slaughtered (pinched) first. Any left over blood will be squeezed out on the altar.

ולא יבדיל [AND HE SHALL NIP ITS HEAD …] BUT SHALL NOT SEPARATE IT — i.e. he nips only one organ (either the gullet or the windpipe) (Chullin 21a).

ולא יבדיל, “he must not sever the head (of the bird) completely. Only one of the vital conduits of the bird, either the one for breathing, or the one for leading to the stomach needs to be pinched.

עורף is the surface of the head which slopes down towards the neck);

‎‏מול עורף is the part bordering on it which sees the עורף (the part next to the עורף and within sight of it) — that is, the length of the whole back of the throat (Sifra, Vayikra Dibbura d'Chovah, Chapter 18 7: Chullin 19b).