So here is a procedure I actually developed for myself couple of months ago. It's self-helpy (the purpose was to solve my self-esteem issues) but I think indignant moralizing uses some of the same mental machinery so it's relevant to the task of becoming less judgemental in general.

I believed that self-esteem doesn't say anything about the actual world so it would be a good idea to disconnect it from external feedback and permanently set to a comfortable level. At some point I realized that this idea was too abstract and I had to be specific to actually change something. And here's roughly what it led to:

1. Notice that I'm engaging in judgement. If the judgement is internally-directed and negative then the trigger will be anxiety. If it were positive then it would be some sort of narcissistic enthusiasm. If the judgement were directed at another person then it could be a feeling of smugness, if negative, and probably some sort of reverential admiration if positive.

2. Realize that the emotions I'm feeling don't represent objective reality. They are a heuristic hacked together by evolution to guide my behaviour in a savannah-dwelling hunter-gatherer tribe. And I'm definitely not currently a member of such a collective.

3. Remember that thinking abstractly about a 'sense of self-esteem' doesn't capture the way it is experienced and that thinking that it should be disconnected from external stimuli isn't something that can be translated into action and I need something specific to target.

4. Focus on how an algorithm feels from the inside -- that the sense of self-esteem doesn't feel like a sense of self-esteem. It feels like a feature of the world. As if everyone, including me, had an inherent, non-specific aura of awesomeness that I were able to directly perceive, though not with any of the 'standard' senses.

5. Reflect on the silliness of that way of perceiveing. Look at the world and notice the distinct lack of worthiness everywhere I turn to. Tell myself, verbally, that there is no inherent awesomeness or worthiness and that therefore nothing can affect it. Don't just try to disconnect the emotions from experience, aim to outright destroy them (note: I don't claim that destroying them is actually possible).