It seems that the part of yourself who judges others does so because they themselves have been judged, and gather resentment towards those who would do such a thing. Where this all starts is the next mystery to be solved.
In terms of interacting with a judgmental person or part of yourself, it seems that they only let down their barriers when you are willing to admit your mistakes and really be humble in front of them. Rather than get pissed at the judgments they throw at you. If you do, a larger and larger fight will brew, and explode, instead. Heck, maybe SOME of the things they say ARE true. To avoid becoming judgmental you must develop, I feel like, the ability to TAKE IT. To take the judgment or criticism, however harsh or mean-spirited. Even if it is designed with the specific intent to crush your spirit.