I'm 15 years old, one of my nipples is puffy & sticks out as the other is flat. What procedure would you recommend to fix this?

I would also not intervene at age 15, particularly for female breasts. Reducing the size/prominence of a nipple is relatively easy. Increasing the size of the nipple is difficult unless it is retracted and can be released. Therefore the simplest procedure would be to reduce the larger nipple to closer to the size of the smaller one but I would wait until more fully grown and get an exam by an experienced plastic surgeon who could discuss options and degree of difficulty as well as your preference for the best nipple size. 

I don't think that touching your nipple has anything to do with one nipple being more prominent than the other. Humans are often asymmetric with one breast larger than the other in most women or one ear higher on the head. At age 15 you have a lot of growing to do and this will probably even out with time, particularly since it does not feel firm, as in true gynecomastia.