Growing full-sized organs

Growing organs in chimeric animals

Probably the most perspective method because you get organs with your own DNA so they won't be rejected.
See The chimaera challenge.
2010. https://pubmed.ncbi.nlm.nih.gov/20813264/ - grew rat pancreas in mouse. "Although full maturation into adulthood was not common, once the mice matured into adulthood, the generated riPSC-derived pancreas was morphologically and histologically normal and was not associated with any sign of diabetes or other abnormalities; GTT results strongly indicated normal function".
2017. https://pubmed.ncbi.nlm.nih.gov/28117444/ - grew mouse pancreas in rat. Pancreas grew rat-size. After that, pancreas were transplanted to diabetic mice (syngenic C57BL/6N). "The transplanted islets successfully normalized and maintained host blood glucose levels for over 370 days in the absence of immunosuppression (excluding the first 5 days after transplant)", "Our immunosuppression regime was stopped after only 5 days, which appears to have allowed enough time for the host (mouse) immune system to eliminate residual donor (rat) cells without triggering a hyper-acute response".
Mouse and rats are separated by ~21M years of evolution, while humans and pigs - by ~90M years[1].
There are also several papers like [2][3] where authors grew mouse lung in mouse that genetically unable to grow lung with the same technique, or pig organs in pigs genetically unable to grow those organs. Interesting but not that impressive as interspecies chimeras.
Overall, while chimeric approach seems to be more perspective than xenotransplantation in the end, there are probably about 2 orders of magnitude less experiments than in xenotransplantation field.

Xenotransplantation

Transplantation of genetically edited (to improve immunocompatibility) animal organs could also solve the problem.
(1) Everyone knows about GMO pig heart to human transplantation, with patient survived for 2 months https://pubmed.ncbi.nlm.nih.gov/35359338/.
(2) https://pubmed.ncbi.nlm.nih.gov/34331749/ (2022y) GMO pig kidneys were transplanted into macaques (with simultaneous nephrectomy). Six macaques survived 15, 20, 71, 135, 265 и 316 days (two other macaques followed slightly different protocol and survived 2 and 61 days).
(3) https://pubmed.ncbi.nlm.nih.gov/35616243/ not really successful experiment with pig kidney transplantation into brain-dead patient: "Although the right kidney graft produced urine initially following the surgery, its actual glomerular filtration rate was minimal, and the amount of urine flow decreased over time. Moreover, the left kidney graft did not work following transplant <...> the team had to terminate the study within 3 days in part due to physiologic instability of the recipient"
Though brain-dead recipient even before transplantation was in a bad health: "The recipient exhibited multiorgan dysfunction such as liver failure and coagulation/fibrinolysis disorder before transplantation". So might be lack of success here doesn't matter much.

However, it's worth to mention that the field growth is not very fast.
(1) As early as 1963, one of 13 patients lived 9 months after a chimpanzee kidney transplant<ref>https://doi.org/10.1016/B978-0-12-398523-1.00069-0 really interesting review, strongly recommend</ref>.
(2) Back in 1999, one baboon lived 39 days with a GMO pig heart (his own heart was cut out). For 38 days he was active and vigorous, and on the 39th he abruptly gave up and died. Post-mortem histopathology did not see active heart rejection, only "small foci of mild humoral rejection". https://pubmed.ncbi.nlm.nih.gov/10703701/
(3) In 2004, one baboon lived 83 days after replacing his kidneys with those of a GMO pig https://pubmed.ncbi.nlm.nih.gov/15619627/ and died for reasons not directly related to the transplant ("grossly and histologically, the renal graft showed no evidence of rejection").
(4) Another 2004 achievement was that one baboon's heart transplanted from a GMO pig lasted six months https://pubmed.ncbi.nlm.nih.gov/15619628/ (the baboon also had its own heart, so the transplanted heart was cut out and the baboon continued to live) (other baboons in the experiment were less lucky, the median survival rate of the transplant was 78 days, some baboons died), and in 2012 the record was increased to 236 days https://pubmed.ncbi.nlm.nih.gov/22070772/

Possible cons:
(1) Pigs live ~15 years. Would their organs live more? or at least 15 years given that conditions in human body are different?

Organoids

https://pubmed.ncbi.nlm.nih.gov/33623712/ Review in Nature, 2021. "(partial) absence of a mesenchymal compartment, vascularization and/or microbiome", "An important drawback of organoid systems is the limited time span for which they can be maintained in culture. Epithelial organoids have lifespans on the order of one week, which is often insufficient to robustly differentiate", "In consequence, these organoids generally fail to mature beyond a fetal phenotype", "As organoids grow in size, diffusion-dependent nutrient supply and waste removal become less efficient. For example, in cystic epithelial organoids, dead cells accumulate in the hollow lumen and, thus, the organoids have to be fragmented and reseeded. In brain organoids, which can grow to several millimetres in size, nutrient inaccessibility leads to necrosis of the inner core, which can only partly be resolved by shaking cultures"

Organs-on-a-chip

Good visualisation to grasp the concept: https://youtu.be/CpkXmtJOH84?t=227 (from 3:50 to 5:15)
Good and useful concept but probably not immediately and directly relevant here.

Michael Levine experiments

They are probably not immediately and directly relevant here though might (perhaps) be useful in regulation of chimera growth.

Other

See also futuristic concept of deep learning based body growing.