From what I've read its made from a mat of "agricultural byproduct" which I think is a mat of compressed and dried cow manure or other fibrous decomposing material that is placed on a dense base of pre-fruiting fungus which then grows a collagen rich stalk through the dung.
If rather than the stalk of the fungus, the roots of fungus were used then it would be a matter of finely controlling the growing environment to keep the fungus growing a thicker and thicker root mat through the growing medium.
Timing and cutting the right layer is next step, followed by sterilizing the material to prevent further growth.
Given the assumptions with the setting its feasible, there might be something like an incredibly expensive early thermometer involved though.
[Edit] I think they have the glass technology required.
https://www.fastcompany.com/40562633...rooms-not-cows