I have the same issue and have 2 solutions.
1) stick a (clean) sock in your mouth and breath through it.
It won't eliminate breathing sounds but it minimizes what the mic pics up and keeps you aware of your breathing. FYI it does not have to be a sock.
2) singers learn to breath and exhale. One of the key concepts in good singing is to keep your soft palet lifted (making a gu sound, as in the word guitar can help connect. Other words such as guy, gay and goo.....help). If you can maintain a lifted soft palet whlle inhaling and exhaling through your mouth it will go a long way to getting rid of the breathing sounds. It takes time and practice. Google lift soft palet and you can find loads of info.