Madame Lil can train you as an expert. I am not sure about the character conditions for this particular line, you may see my character stats in attached files. Note that I also applied latest Unofficial Arcanum Patch before playing (link below). Perhaps it's important to be not very evil, have at least INT 8 and and point in Haggle, or maybe reaction >39 is more than enough.

During my playthroughs I just complete first 2 or 3 delivery quests for her, train my Persuade up to 3 points and then proceed for the Expert training. Afterwards she'll tell you where you can find a Master of Persuasion.

As for the Medallion of Beauty is the 4th delivery quest, completely unrelated to that line.

Link:
http://terra-arcanum.com/drog/uap.html#downloads

good to know, Cadaver !

You do not need to find the medallion; as Cadaver747 says, that's an unrelated quest.

You do, however, need to already be an apprentice. Did you request apprenticeship training from one of her girls first?

If so, the "I would like to be trained as a Persuasion Expert" should be in the first dialog window as soon as you first talk with her. If it's not there, then something else has gone wrong. Verify that you do have Persuasion apprenticeship; and if so, I'd suspect a corrupted save file. (Which can happen very easily, if you're in the habit of over-relying on F7 and F8 to quicksave and quickload.)