The person sulking is doing so because they dont feel good about their side of things. I'm going to say whether they are intentionally sulking or not is irrelevant. The bottom line is that they don't feel good about it.
This person should take some time to think about why they feel bad and what might make them feel better. (Not focusing the big main topic of contention, but on smaller more accessible things).
The person making the request should try to do what they can to help the other person feel more (whatever it is they are craving) - more secure, more loved, more safe, etc.....
These sorts of things usually come down to a sort of insecurity at their roots. So figuring out a way to reassure the particular insecurity can really help.
I genuinely wish you all the best.
If you have access to it, I highly recommend SARK's "wild succulent love" - she talks a lot about getting to the core of issues to resolve them
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...what do the public, the great unobservant public, who could hardly tell a weaver by his tooth or a compositor by his left thumb, care about the finer shades of analysis and deduction!