There have been several threads on this including the recent Falling for a Provider series, so if you are an active reader, I suggest you look at them. In my cases, I have usually asked the girl out after several visits and our conversation during the sessions has gotten to the level where we are talking about our interests, etc. I have had several relationships/friendships with AMP ladies, and they are fun but hard. Usually they end because we just don't have enough in common, and there is difficulty in finding a good time to be together since evenings are usually their busiest work hours. I have also had cases where the girl is just looking for a friend away from work and doesn't want you to be a sex partner because they save that for lovers.
You need to be clear on what your motives are. If you want to attempt to build a real relationship, you have to be prepared to accept what she does, figure out how you will introduce her to other friends and answer the question on how you met if your friends ask, etc. If you don't want to attempt a relationship, you are probably just hoping to get a better deal by taking the relationship outside. That may lead to problems as she may have expectations that you never intended, or she may be disappointed that you are still treating her as just a sex partner.
One thing to worry about. After becoming friends and seeing one away from work, they have not wanted me to continue as a customer. So if the relationship breaks up and she was a favorite provider beforehand, you may be in a situation where you can't even purchase services from her again.
Good luck.