The customary practice when reading Parashat Ki-Tisa is to end the first aliya just before the story of the sin of the golden calf (31:17), and to then read the entire narrative of this story for the second aliya. This results in a very lopsided distribution of the parasha – featuring two unusually lengthy aliyot followed by five relatively brief aliyot.
This practice is mentioned already by Rav Chaim Benveniste, in his Kenesset Ha-gedola, as cited by the Magen Avraham (428:8). The reason this is done is so that the story of the golden calf would be read entirely by a Levi, who receives the second aliya. As Levi was the only tribe which did not participate in the sin of the golden calf (as indicated in 32:36), it is appropriate to have this narrative read specifically by a Levi. (Interestingly, the Kenesset Ha-gedola also writes that it is customary to read this section in a low, subdued tone, a practice which is not commonly observed nowadays.)
Rav Chaim Palagi, in his Yafeh La-leiv (428:3), writes that if the kohen who receives the first aliya wants to continue and read also the story of the golden calf, he may be allowed to do so. Since kohanim also belong to the tribe of Levi, the custom to have a Levi read this section can be fulfilled also by a kohen. (Of course, the remainder of the parasha must then be divided into six aliyot, instead of the usual five.) However, Rav Palagi then cites those who disagree, arguing that since kohanim are the descendants of Aharon, the one who actually made the golden calf (notwithstanding the fact that his intentions, according to the traditional understanding, were sincere, hoping to stall until Moshe returned), it would be inappropriate for a kohen to read this section.
Rav Palagi’s ruling might perhaps serve as a precedent for a similar case – where no kohen is present in the synagogue on Shabbat Parashat Ki-Tisa, such that the first aliya can be given to any congregant. It stands to reason that if a Levi is present, it would be preferable to call the Levi for the first aliya, and extend the first aliya until the end of the story of the golden calf, so that this section is read by a Levi – similar to the kohen in the case discussed by Rav Chaim Palagi. Indeed, Rav Yitzchak Zilberstein, in Chashukei Chemed (Menachot 30a), tells of a synagogue in which this was done when no kohen was present on Shabbat Parashat Ki-Tisa, and Rav Zilberstein’s brother-in-law, Rav Chaim Kanievsky, said that this was the correct procedure to follow under the circumstances.
Rav Zilberstein addresses also a different case that once happened in a certain synagogue – where neither a kohen nor a Levi was present, and a Yisrael was called for the first aliya, but right when the reading began, a Levi walked in. Once a Yisrael is called for an aliya, no subsequent aliyot can then be given to a kohen or a Levi, and thus, seemingly, in this situation, there is no possibility of following the custom to have a Levi read the story of the golden calf, since a Yisrael was already called for the first aliya. However, the gabbai at this synagogue conceived of a creative solution. Halacha permits adding aliyot on Shabbat beyond the seven required aliyot, and Leviyim may be called for the additional aliyot. The gabbai thus suggested that the first section of Parashat Ki-Tisa, which is normally read as the first aliya, should be divided into seven aliyot, and then the Levi who had just arrived can be called for an eighth aliya, which would extend from the beginning of the story of the golden calf through the end of the parasha. This way, a Levi will be able to read the story of the golden calf. Of course, the result of this arrangement is an extremely imbalanced distribution of aliyot – seven very short aliyot, followed by one exceptionally long aliya.
Rav Zilberstein writes that in principle, this arrangement should be followed in such a case, in order to fulfill the custom of having the story of the golden calf read by a Levi. However, the leaders of the congregation must determine if whether having such an exceptionally long aliya will cause the congregants to grow impatient and restless, in which case it is preferable to conduct the reading as usual, as the custom to have the story of the golden calf read by a Levi does not justify overburdening the congregation.