Your question touches on several halachic issues, which we will mention only in passing as necessary background for the answer to your question.
As is well known, it is necessary to remove the blood from meat before one may eat it. In truth, however, not all blood is the same; the main halachic issue relates to blood that moved from its initial position.[1] Salting, following the regular procedure, which includes rinsing the meat, is usually sufficient to remove the blood.[2] However, due to the high concentration of blood in the liver, salting is insufficient to remove the blood from the liver meat, and the more powerful process of broiling is required, after slitting the meat to allow its blood to flow out more easily.[3] (There are important halachot regarding this process that we assume you either know or will learn before carrying it out.)
One of the situations in which it is more difficult to remove blood from any meat is when the meat remained unsalted for 72 hours after shechita.[4] In such a situation, it is possible to halachically remove the blood only through broiling the meat,[5] but the classical poskim differ significantly as to the extent to which broiling is fully effective in that case. The Shulchan Aruch[6] takes a middle approach. On the one hand, he is concerned that broiling after a 72-hour delay might not remove all the blood, such that if one cooks (or fries or sautés) the meat afterward, some blood might still come out and render the meat not kosher. Therefore, l’chatchila, one should not cook such meat even after broiling it. On the other hand, the Shulchan Aruch rules that since it is not certain that further blood will be displaced in the process, b’dieved, if one did cook such meat after broiling, it may be eaten afterwards.
Many decades ago, a major halachic dispute emerged regarding the halacha when meat is left unsalted for 72 hours in a frozen state. (Nowadays, the salting process is almost always performed at the slaughtering facility soon after the shechita, before the meat is frozen.) Some significant poskim[7] reason that if one freezes the meat solid (basar kafu), so that chemical processes are suspended, the “72-hour clock” stops. Others, however, argue that freezing cannot change the halacha. If one adopts the lenient approach in this dispute, broiling should suffice in a parallel case regarding liver as well, assuming it was frozen well within 72 hours. Your question arises only if one follows the stringent opinion regarding salting frozen meat.
Based on the above, we can understand the following ruling of the Pitchei Teshuva.[8] He cites the Chamudei Daniel as maintaining that although one should not let meat sit for 72 hours without kashering it, because he might come to improperly rely on salting instead of broiling,[9] one may let liver sit that long, because in any event one must always broil liver to kasher it. He notes that broiling the liver is not a full solution for liver that sat 72 hours (which is no better than meat that went unattended for such a period), since one is not allowed to cook it after broiling. However, since we rule that if one did cook liver after a delayed broiling, it may still be eaten after the fact, the issue is clearly not such a serious one. We may therefore allow liver to sit 72 hours before being kashered.
The same approach of relative leniency regarding liver that is to be broiled, as compared to meat that is to be salted, will help answer your question as well. It is a significant stringency to keep the 72-hour clock ticking when meat is frozen. Therefore, it is certainly halachically acceptable to use liver freely after broiling it, even if it had first been frozen for 72 hours.[10] As Rav Ovadia Yosef[11] explains cogently, it is also permitted l’chatchila to set up such a situation by freezing the liver for an extended period of time and then broiling it.
There is a difference of opinion among poskim regarding whether one must wait for the liver to thaw before broiling it (so that the beginning of the process will not be considered cooking the liver in liquid as it thaws) or whether broiling is effective in any case.[12]