Okay, I figured out a reasonable way to do it…
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Have an input for the hour only:

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When that input’s value is changed, run this action:

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Repeat steps 1 and 2 for the minute.
If you’re using 24-hour time, you can stop there. If you want to use 12-hour time, I was hoping I could make an am/pm dropdown box, but I don’t know how to then use that to update the time. Any ideas?
Edit: The above solution ended up having a couple of issues. Discussion continued here: How to allow users to edit a time? - #5 by emasl