I wouldn't launch into something incredibly daring right off the bat, especially if it is a legitimate stuttering problem and not just approach anxiety. Legitimate stuttering problems are best dealt with by going to see a professional, rather than throwing yourself into an embarrassing situation just to try to learn to not care about being embarrassed. This could actually make it worse, if, for example, you went on stage to do some public speaking and performed extremely poorly. You might develop an even greater fear of that type of situation because you associate that extreme and unpleasant embarrassment with public speaking, even though no real harm came to you. Stuttering problems can be rooted in deeper psychological layers than normal approach anxiety, and other types of therapy besides desensitization may be best to try first.
I would start small and work your way up. If this has deep psychiatric roots (overbearing and unempathetic fathers are a pretty common source of stuttering) then you should see a professional. If it only occurs in a few select situations and you think you can overcome it just through practice and your own courage, then start small. Practice reciting difficult passages from a religious text or tongue twisters in the mirror over and over. Try singing when you are by yourself, and correcting your tone of voice to try and get yourself to actually sound good when you sing. If you have any friends that you trust to be there for you when you need them (more for your comfort rather than them, if they wouldn't support you in this then they aren't worthy of being your friend anyways) then talk to them about this and ask for their help. If they don't have approach anxiety, ask them to introduce you to people as practice or you could just ask them to do mock conversations with you where they roleplay as a stranger. Go out to some place you wouldn't normally go, where people you wont see in other parts of your life will be, and just walk up to them and ask them questions or just try to introduce yourself, knowing that you will never see them again. You can even walk up to people and say "I am introducing myself to you in order to overcome my stuttering."
Just remind yourself that you never have to see them again. Many people will even admire your courage and dedication if you walk up to them and say this.
Another thing that is very helpful in social situations is learning how to eject yourself from a conversation properly and at the right moment. Learn to anticipate fatal awkward silences (some silences are shorter and not fatal to a conversation if you insert nonverbal communication) and eject when you know they are coming and you can't think of a way to stop them. Ejecting should not be a first resort, but it should come naturally when you know that you WILL NOT be able to save the conversation because you know that paralyzing embarrassment is 3 seconds away. You have to get good at anticipating this (which of course means seeing it happen a lot). Come up with some situations where you would be talking to someone, then come up with some natural ways to walk off feeling like you left behind a good impression. These shouldn't be excuses, but more like "I'm an important busy person and I have to go" type of things. For example, when I go to the dog park and I approach girls there, if the conversation is stale or the girl is boring, I make enough small talk with her to keep it natural, and then eject immediately right before it gets awkward. I simply just say something like "that's my cue, see you around" when my dog lies down and walk off. If I don't want to leave and there are other people to talk to, ejecting is also a great opportunity to launch into meeting someone else. You can simply casually say "nice to meet you" and walk over to someone else (which is perfectly fine, trust me). This has the double benefit of getting you out of a bad conversation and forcing you to ignore your anxiety and start a new conversation with someone else. Once you are comfortable with the simple idea that you can bail on any conversation politely with an air of confidence, you feel much more at ease being in the conversation because you know that you can avoid whatever super awkward or embarrassing thing it is that haunts your dreams. Ultimately, your conversations go better, last longer, and you walk away from them feeling good simply because you are more at ease knowing you could leave if you wanted.
Also, work on body language. I think people undervalue body language by simply marketing it as a way to convey feelings to the other person. Having relaxed and comfortable body language is also beneficial because it actually forces you to feel relaxed and comfortable after a while. Even if you are initially uncomfortable in a conversation and displaying traditional relaxed body language doesn't feel "right," simply forcing yourself into that body language and keeping it there can knock your anxiety down a few pegs.