Should I Extract Two Upper Premolars or Can I Go Without Extractions? (photo)

Don't extract the teeth, you will be extremely unsatisfied with the results. Extracting two upper bicuspid teeth and retracting your upper front teeth to match your lower teeth is a terrible idea. Your problem is with the lower jaw, it is overly retruded. You need to bring your lower jaw forward. This can be done with a simple appliance, NOT surgery. However, the reason your lower jaw is retruded is because your upper arch is underdeveloped and therefore blocking out your lower jaw. 

First you need to develop the upper jaw (SLOWLY, NOT rapid palatal expansion). Then you will have enough space to straighten your upper teeth. Then you need to bring the lower jaw forward to match the well-prepared upper arch.

Good luck!

I can't recommend whether or not to extract teeth without seeing clinical photographs or performing a clinical examination.  What I can tell you is that to relieve 9-10 mm of overjet in an adult without spacing present, would require tooth extractions, or even orthognathic surgery (in the case of a retrognathic mandible).   In terms of "developing the upper jaw" to straighten the upper teeth as in the cosmetic dentists response to your question, I would not count on non-surgical expansion of the upper jaw to be a stable treatment, nor would it relieve your overjet.  That being said, if your goals are simply to align the teeth without resolving the overjet, it may be possible without extractions.  Good luck!