What wolfens linked to is a valid list, but there are as many professions as you can think up.
CoE doesn't have professions in the traditional sense so much as a set of skills that you can use to do various things.
If you want to learn animal husbandry, and pharmaceuticals, and then some first aid too, you could feasibly call yourself a veterinarian even though that profession doesn't exist on a list anywhere.
What you need to do for Falconry is determine if Falcons exist, if you can train them, and then if you can call this your profession. The answer to the last one at least, being yes.
Likewise if you want to make any other profession a thing, consider what that profession does and research if the game mechanics will allow you to do these separate skills. An example of the most recent skill "flower" being this.