Do not — as a British journalist did some years ago — confuse this useful word with pogonotomy. Both originate in the Greek word pogon, a beard, but the latter ends in –tomia, cutting, and so is the word for trimming one’s beard, or shaving, the exact opposite of the writer’s intention.
Pogonotrophy, on the other hand, ends with Greek trophe, nourishment, so its literal sense is “beard feeding”, though it it may better be glossed as growing a beard or cultivating one.
Sam Brinkley was perhaps the world’s greatest authority on pogonotrophy. That’s right, pogonotrophy, the science of beard growing. Sam didn't start out to be an authority. As a young man he simply became disgusted with the speed with which his beard grew. Twice a day he had to shave the stubble from his face. So, in his early twenties he got mad and quit shaving, his wife’s protests to the contrary. His beard grew and grew — down to his chest, down to his navel and down to his waist.
Scoundrels, Rogues and Heroes of the Old North State, by Dr H G Jones, 2007.
Other words in pogon include pogonic, pertaining to a beard, and Pogonophora, the systematic name for a group of deep-sea worms; their name actually means “beard bearer”, which is odd, since they don’t have mouths to cultivate them around.