What is Santa's beard made from?
It's difficult to know for sure, but it looks to me like plaster of Paris, also known sometimes as craft plaster when sold in craft stores, which is basically just gypsum plaster. Since it is real plaster, it would be fairly brittle and prone to breaking, especially when applied in a thin layer, as it seems to be on your Santa. It also generally dries white, which would explain the lack of brush stroke on the beard - no paint seems to have been applied.

I've never tried adding plaster on top of already dried plaster, so I don't know if it would stick well. Plaster of Paris also generally comes in packages of at least one pound of powder, so if patching the beard is all you would do with it, it might not be worth it. Drywall compound would likely work fine for patching and might not end up all that different, since it is also made primarily of gypsum powder.