Since I haven't found any posts about it, I figure we're missing something obvious as usual. How is this card supposed to be used?
Yeah, I think you're overthinking it. Mining says "One-time Bonus of 1 production per plot".
So, at the start of your next turn, when you collect your resources that your plots give you, count your plots and take that many more in blue production cards. 5 plots = 5 more production, etc. Next turn, just take the normal amount, since it's one-time use.