According to the short description of a "Falconry Master" that Fawn could read below its name, it was a profession that focused in birds of prey, teaching them, learning from them as well as training them and with them.
The word "master" in the name of the profession didn't wasn't used in the owner or superior sense, but as a sign of the greater potential the person in question had in the respective profession when compared to the other options, simply put if a profession with the word "master" appeared, it was because the blue flaming board judged one to be the most compatible with it.
Of course, at the end of the day, one's own thoughts are what mattered the most, that's why Aster told the girls to chose what they wanted and felt the closest to, which so far has shown good results.
