Shalom Aleichem, Rebbi u'Mori!

Thank you for your question. The Sugya you are asking about is very difficult to understand just by reading through the subject, and there a lot of opinions. I will try to bring the main ones.

I. The Gemara seems to say that there are two options: either the Torah was given chapter by chapter, or the Torah was given as one book. There are several opinions how to explain this.

1) Rebbi Yochanan (chapter by chapter):

a) Rashi says that according to Rebbi Yochanan, the Torah was written chapter by chapter, every time a new story or Mitzvah was given to Moshe.

b) The Ritava, Rashba, and, later on, the Mabit (Beis Elokim 24) say that the reason the Torah was given like this is because this way it is easier to teach the Torah to the Jews.

c) The Mabit adds that some say that Moshe wrote it all at one time but did not present it yet to the Jews (besides the eight Parshiyos which the Gemara brings). He says a great Chidush, too -- that whatever was not given to the Jews in writing, they were not punished for if they did not fulfill those Mitzvos.

d) Some say that according to Rashi, the parts of the Torah before Moshe were written first by the heads of each generation (Adam, Noach, Avraham, etc.), and these are the Megilos to which Rebbi Yochanan is referring Others (such as the Re'em) argue and say that Rashi agrees with the Ritva and Rashba and he, too, means that Moshe wrote those Megilos as well.

2) Reish Lakish (one book):

a) Rashi says that nothing at all was written before the end of the forty years in the desert, and only then did Moshe write the whole Torah from his memory.

b) Tosfos says that Moshe did write the Torah also according to Reish Lakish, and the argument between Rebbi Yochanan and Reish Lakish is only in cases where the Parshah was given to him out of order. According to Rebbi Yochanan, he wrote each Parshah seperately, and in the end he put them in the order they are today. Reish Lakish argues and says that in a case of Mukdam u'Me'uchar, he waited until the later Parshah -- that needed to be inserted out of the chronological order -- was told to him, and then he wrote them in the Torah order.

It is important to add the discussion in the Rishonim, who question Rashi's explanation, especially his understanding of Reish Lakish, that nothing at all was written until the end of the forty years in the desert. In Parshas Mishpatim, the Torah specifically says that Moshe read from the Sefer ha'Bris, so what is that Sefer, especially according to Reish Lakish? It is also a question on Rebbi Yochanan, who says there was no Sefer Torah yet, but just separate chapters.

II. There are several explanations what this "Sefer ha'Bris" was, and when it was written. I'll bring a few explanatins according to the two opinions of Rebbi Yochanan and Reish Lakish.

1) According to Rebbi Yochanan:

a) The Ramban in the preface to his commentary on the Torah says that Moshe actually wrote the Torah from the beginning until after the Parshiyos of the Mishkan at the end of Sefer Shemos, and the reading of this Sefer is later than where it is inserted in the Torah.

b) The Ramban says there that there is another opinion in the Mechilta, that he wrote this Sefer ha'Bris on the day he came down from the mountain, and not after the commandment and building of the Mishkan.

c) Other Tana'im in the Mechilta say that he wrote this Sefer on the Thursday before Matan Torah, which was on Shabbos.

d) According to Rashi in our Sugya, it seems that what Moshe wrote as Sefer ha'Bris was a separate Sefer, and not part of the Megilos which were later on sewed together.

2) Accprding to Reish Lakish:

a) According to Rashi and other Rishonim who say that nothong at all was written before the end of the forty years, the Gur Aryeh and others want to answer that Rashi agrees that Moshe did write some of the Torah before, but it did not have the Kedushah of a Sefer Torah, but just the information was written down. Later, Moshe wrote it all again and the whole Torah became holy with the holiness of a Sefer Torah.

b) Tosfos says that Moshe wrote the Sefer ha'Bris, but then wrote again the whole Torah at the end of the forty years in the desert.

I hope this helps!

Kol Tuv,

Aharon Steiner