In the UK: if you ever visit York, go to Barley Hall. It's a surviving 15th century town house with interior furnishings mostly reconstructed according to period accurate methods and styles. It's a little out of the usual tourist trail and is much less well known than the Minster or Jorvik, but definitely worth an hour or two.
In London, I wouldn't exactly call the V&A 'lesser known' but it has much better medieval European collections than the more popular British Museum (and is always far less busy). The Courtauld Institute over near the Strand also has some nice medieval art.
If you're in Canterbury (which you should definitely do if you're visiting the UK and like the middle ages), the 'Canterbury Tales' attraction is a mix of animatronics and live actors retelling parts of Chaucers tales. It's goofy as all fuck, but every single one of my Middle English Literature colleagues who visited it loved it.
As for the US: I'm not sure either if it really counts as lesser known, but the Met Cloisters in NYC is easily in my top 3 favourite museums in the entire world and is really worth going a bit further out of the usual tourist areas in the city.
I'm interested to see what everyone else recommends here? I'm sure there's so much I still haven't seen in the UK. (I've also not counted churches or castles in this list, because that's a whole different ball game)
The Wallace Collection in London has an excellent collection of medieval and renaissance weapons and armour, alongside its other exhibits. I have never been personally but I have used the items on their website for research while a university and have had recommendations to visit. IIRC the curator of arms and armour has written extensively on the topic and is considered one of the leading experts.