Commentaire sur Le Lévitique 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.