Unless the other person is just being a huge dick, the best way to handle someone mocking you is to not be defensive, it shows the other person what they are doing it something you are self conscience about. Show confidence, act like you dont care, play along with them, laugh at what they said, maybe take a small jab back at them if it feels appropriate. Growing up with older siblings has taught me how to handle those situations and I promise you it is almost always better to get humorous/ mildly sarcastic with the other person then to get angry. That anger shows weakness and it will further damage your self image. To overcome stuttering you have to not be so insecure about it! trust me i know its hard, I stutter too!