I apologize in advance, a lot of the papers referenced may be paywalled.
There's a lot of literature on the various Herpesviridae and it is a large pleotropic family, producing such viruses as Cytomegalovirus (CMV), Human alphaherpevirus-3(AKA Varicella-Zoster virus (VZV)/chickenpox/shingles), Human alphaherpesvirus (AKA Herpes simplex virus(HSV)/cold sores), Human herpevirus-6 (roseola/sixth disease). Obviously, many of these do not have any skin involvement at all, and so won't produce scarring. However, ZVZ in both its chickenpox and shingles forms does scar 1 (I can also personally attest to the chickenpox bit, no shingles yet), and HSV also scars 2, but often only after repeat occurrences at the same location and is a common cause of ocular scars with vision loss 3.
The genomes of the Herpesviridae are large and not fully studied. They are about 100,000- 250,000 base-pairs and contain quite a number of genes, generally in the 50-100 range, with a number of proteins being produced from secondary processing of the transcribed RNA (splicing etc). Not all proteins are fully characterized in any of the species that I can see. I have found a fairly complete description of the gene products and proteins for CMV in Van Damme and Van Loock4 (see supplementary PDF), and Cohen5 and Zerboni et al6 for HSV, and there are a number of immune modulators and apoptosis inhibitors that affect how the cell or body respond to infection.
Most notably is an interleukin 10 (IL10) homologue, vIL10 (protein UL111A in CMV), which is also found in a couple of other Herpesviridae members, but not in VZV or HSV as far as I can tell. IL10 is implicated in wound healing, but is also used for a bunch of immune response maturation processes. In addition, there are a number of proteins produced that may play roles in suppressing immune responses and activate growth factors, but are not easily identifiable as they lack structural and sequence homology to currently known ones. The current state of things for VZV is nicely reviewed in Zerboni6, linked above, but is more focussed on how these factors affect viral output rather than healing post-infection
Edited to add: One group of authors claim to have identified some fractions from cells infected with HSV that have growth-factor-like properties7, which have been sporadically looked at and published since the 1990s. In these it seems that the fractions probably contain several viral proteins that may contribute to the growth factor function, but as of 2016 no further work has been done on them. I had initially ignored these as they seemed to have not clarified exactly what was going on, but the research seems good.
1 Moffat J, Ku CC, Zerboni L, et al. VZV: pathogenesis and the disease consequences of primary infection. In: Arvin A, Campadelli-Fiume G, Mocarski E, et al., editors. Human Herpesviruses: Biology, Therapy, and Immunoprophylaxis. Cambridge: Cambridge University Press; 2007. Chapter 37. Available from: https://www.ncbi.nlm.nih.gov/books/NBK47382/
2 LAWRENCE COREY, HARRY G. ADAMS, ZANE A. BROWN, et al; Genital Herpes Simplex Virus Infections: Clinical Manifestations, Course, and Complications. Ann Intern Med.1983;98:958-972. doi:10.7326/0003-4819-98-6-958
3 Whitley R, Kimberlin DW, Prober CG. Pathogenesis and disease. In: Arvin A, Campadelli-Fiume G, Mocarski E, et al., editors. Human Herpesviruses: Biology, Therapy, and Immunoprophylaxis. Cambridge: Cambridge University Press; 2007. Chapter 32. Available from: https://www.ncbi.nlm.nih.gov/books/NBK47449/
4 Van Damme, E., & Van Loock, M. (2014). Functional annotation of human cytomegalovirus gene products: an update. Frontiers in microbiology, 5, 218. https://doi.org/10.3389/fmicb.2014.00218
5 Cohen J. I. (2010). The varicella-zoster virus genome. Current topics in microbiology and immunology, 342, 1–14. https://doi.org/10.1007/82_2010_10
6 Zerboni, L., Sen, N., Oliver, S. L., & Arvin, A. M. (2014). Molecular mechanisms of varicella zoster virus pathogenesis. Nature reviews. Microbiology, 12(3), 197–210. https://doi.org/10.1038/nrmicro3215
7 Lachová V, Svetlíková D, Golais F, Šupolíková M, Turianová L, Betáková T. Transforming Activity of Murine Herpesvirus 68 Putative Growth Factor Is Related to the Ability to Change Cytoskeletal Structure. Intervirology. 2016;59(3):137-142. doi: 10.1159/000453067. Epub 2017 Jan 5. PMID: 28052265.