Can you use a frozen loaf of bread for lechem mishna when you make hamotzi over challah on Shabbat?
On Shabbat, it is a custom to say the blessing for bread (hamotzi) over two loaves of bread (lechem mishnah). The Gemara in Brachot says that a person is obligated on Shabbat to make hamotzi on lechem mishnah. Rav Kahana had a custom to take two loaves of bread, but only rip pieces off of one of the loaves. Rav Zera on the other hand, would rip pieces from both loaves of bread. Rav Hai Gaon and others hold that one can follow either custom.
From the Shulchan Aruch, we learn in סימן רע"ד סעיף א that a person should rip pieces from only one of the loaves, and not both. In Chapter 32 (בשו"ת יביע אומר סימן לב) it says that this is also stated in the Kabbalah. Since the Shulchan Aruch says that a person should rip pieces from only one of the loaves of bread, the second loaf can be frozen. Despite the fact that this frozen bread is currently inedible, it will be edible as soon as it defrosts. Therefore, it can be used for lechem mishnah.
Even though the obligation of lechem mishnah is not a biblical obligation, we get the concept of lechem mishneh from the pesukim in the Torah (what is usually called אסמכתא).
The Chazon Ish and the Response of Peri Hatazadeh (שו"ת פרי השדה ( ח"ב סימן פ"ח) ובחזון איש חלק א' קפח) state that we are allowed to use matza for lechem mishnah on Shabbat erev Pesach — even though it is forbidden to eat matza on that day. We hold like Rav Kahana: a person only needs to tear pieces off of one of the loaves. Just as on Shabbat erev Pesach we can use matza for lechem mishnah, we should be able to use frozen bread for lechem mishnah. In both cases, the second loaf is not eaten, yet it can still be used as lechem mishnah.
Furthermore, the Pre Megadim (פרי מגדים על ט״ז א״ח רעד:ב) states that if a person has a custom not to eat bread baked by a gentile (פת עכו"ם), but they have no other bread to use as lechem mishnah, they are allowed to use it for lechem mishnah, although they wouldn’t eat it.
However, both of these examples are referring to something that could be eaten, but in our case, the frozen loaf cannot be eaten.
In ממשניות טהרות פ"ג מ”א, it says that impure liquids do not become pure after becoming a solid. Therefore, bread that becomes frozen is still bread. It can be used for lechem mishna. However, we can also find a difference between this case and the case in our question. Liquids are eaten by some whether they are in liquid or solid states of matter. On the other hand, bread is normally not eaten while frozen. Thus the comparison really isn't valid.
Rabbi Yitzchak Yaakov Weiss (שו"ת מנחת יצחק (ח"ט סימן מב)) says you can be lenient, because some people do eat frozen bread. Besides, the bread could be thawed on Shabbat in an acceptable manner by placing it on a warm pot. In שו"ת אור לציון (ח"ב פכ"א תשובה ב'), it says since we can take the hafrashat challah from frozen baked bread, the frozen loaf is considered bread and acceptable for lechem mishna.
In הגרש"ז אורבך (שש"כ פנ"ה הערה לא), it says frozen bread cannot be used for lechem mishnah. However, if the person has in mind to have a long meal, giving time for the bread to defrost, then they can use it for lechem mishnah.
However, in שו"ת שבט הלוי ( ח"ו סימן לא אות א'), it says that in order to have a proper lechem mishnah, both loaves of bread need to be defrosted.
Finally, in מרן הגרע"י זצ"ל בשו"ת יביע אומר (חלק ח' סימן לב), it says that since lechem mishnah is D'Rabanan. so you can be lenient. Therefore, one can use frozen bread for lechem mishnah. Nevertheless, if one could easily ask their neighbor to give them a loaf of non-frozen bread for lechem mishnah, they will receive a blessing from Hashem.