I hae the same issue! I have a feeling it would be fine if Lightroom were able to open directly on the external monitor, but my laptop is display 1 by diefault and i do not intend to change this just for this issue (as it will gnerate a whole other set of issues for me..).
Sooo Adobe, please please find a fix or help us fix it 😄
I had a similar problem and fixed it by disabling the high DPI for the app. In windows on the task bar, right click on the app and go to properties, then go to the compatibility tab and click "change high dpi settings." Then check "high dpi scaling override," okay it, and restart the app. It should look right after that.
This is the fix! The scaling override fixed the small menus for me. After checking the box, I pulled down the selection below and chose "Scaling performed by System" which gave the properly sized menus on my second screen. Thanks!