What is the recommended OS for compiling a Windows binary for Litecoin?
The Litecoin source tree, in the doc/ folder, provides a file regarding building on Windows, using MinGW and MSYS. I can't personally vouch for those instructions, never having used them myself, so if you go that route, good luck. The way that the release binaries are built, if I'm not mistaken, is using the Gitian builder. If you're looking to do a Windows build, I would try that. If you want to actually use Gitian, you can clone the repo, and use the readme file to get Gitian set up, and then follow the build segment of the release process to do the actual build. On the other hand, if you're not necessarily looking for determinism or the ability to reproduce the exact build, you could try taking a look at the Gitian build scripts to see what packages are being used, what commands are run, et cetera, and try to reproduce that on a Linux system.