The Torah (Shemot 16; Parashat BeShalahh) details the miracle of the mahn ("manna"), which HaShem brought down from the Heavens to sustain 'Am Yisra'el in their journey through the desert.
In a shi'ur of his, HaRav Zamir Kohen SheLIT"A explains that we eat two hallot/pitot during Shabbat meals in order to commemorate the miracle of the mahn that HaShem performed (specifically, that a double portion was provided on Friday to sustain us throughout Shabbat).
This is brought down as Halakhah in Mishneh Torah (Hilkhot Shabbat 30:9), Rabbeinu Asher on Masekhet Shabbat 117B, Halakhot HaRI"F (Shabbat, 16:132B), Shulhhan Arukh (Orahh Hayim 274:1), Mishnah Berurah (Orahh Hayim 274:1, Note 1), and Qitzur Shulhhan Arukh - Yalqut Yosef (Orahh Hayim 274:4). The matter is discussed in the Talmud Bavli, Masekhet Shabbat 117B (near the end of the page).
If that is the case, since HaShem brought down two portions of mahn on Friday (Shemot 16:5, ibid. 16:22), why don't we simply have Lehhem Mishneh for the first of our three Shabbat meals? Why must we commemorate the double portion of mahn at every meal when the double portion itself only arrived on Friday?