Is it possible to transfer fat from my left breast to my right breast?

No, tissue from one breast cannot be transferred to the other because it involves more than just fat. Actual breast tissue that makes milk will not survive the transfer.

Erase an interesting question. Typically fat harvested from Liposuction of the Breast is not used for fat transfer due to theoretical risks of lack of screening for breast cancer. I suppose that risk while mostly theoretical is somewhat illuminated if the fat grafting is back to the breast.
Fat transfer requires substantially more fat than most people believe it so you would probably need liposuction of some other area as well.
Weather liposuction of your breast with fat transfer is appropriate or not is a little hard to say and some of it has to do with what your breast look like.
If your goal is simply to change size and volume and both Liposuction and fat transfer may be highly appropriate.
Breast fat transfer is good at increasing breast volume by one half to maximum one cup with a single round of fat transfer. The procedure does not change the shape of the breast only increases volume.
For women who want to change the shape of their breast different procedures are more appropriate.
If your breast have different shapes then different procedures are probably better options.
I suggest having a few in person consultations with local board-certified plastic surgeons who have substantial experience with breast surgery. Delivering consistent quality results with both Liposuction and fat transfer is more difficult than people believe.
Best,
Mats Hagstrom,MD