I had one that broke off last year during the summer; I just placed it into a new pot, ended up with lots of blooms. As mine broke off I am not sure about a cutting as it as mostly dry before I placed it into the new pot. For me the bottom end puts out branches that jut out from the side and do not look good but the tips seem to do the best. What ever you decide to do should be fine as I did not see any problems with cuttings. When that branch started to go did it brake or start to detach itself from the trunk. If it did you should have cut it off and put it into another pot. Anytime I see one branch that is having problems I just remove them and place them into a new pot. So far this has worked well for me. Have fun and enjoy your plumeria.

I would enjoy the last of your blooms before making your cutting. Yes, new branches will grow from the point where you make your cut on the main plant. You can cut it back very significantly and new branches will still form - no worries there.

With your cutting, clip off the leaves with a knife/scissors so it won't lose its stored water (the leaf stems will fall off on their own in a few days). dip the end in some rooting hormone if you have it (not necessary if you don't), and let it sit in a dry shady spot for about 10 days to let the cut callus a bit. Then you can follow the normal instructions for rooting a cutting (plumeria101.com, etc.)

Oh, and you wouldn't expect to see more flowers this summer even if you left the branch attached to your plant.