I have a language server implementation for my programming language. Since it has been developed much later than its compiler, no stuff like query-based redesign was viable to implement. So, I just adopted the compiler codebase for language server needs in a tricky way: on each document change I just redo parsing, but not other compilation steps and use parsing results to perform fetches from name tables built previously - for some older document state. This state is costly to update and it's done only once in a while and only if input document is syntactically-correct. This approach works surprisingly well (it's fast and completion is mostly correct) and doesn't require throwing the whole compiler code away.

Even more, I can't imagine how people write language servers with full reprocessing on each document change (even with query-based approach). How do they deal with broken syntax? isn't it generally impossible to recovery syntactic errors, so that the end result is as good as expected?

I know Haskell’s LSP years ago just stopped on broken syntax, so the only diagnostic would be the syntax error and you would lose code actions. That was annoying, especially because the LSP was slow.

JetBrains (not an LSP) works up to the syntax error, then recovers and works after. In some cases that are not hard to trigger (e.g. after a closing delimiter), the recovery is wrong and everything after is messed up. But in practice that’s rarely an issue; if I want those later diagnostics and code actions, I fix the delimiters, and the analysis updates quickly.