tsserver provides support for compile on save https://github.com/Microsoft/TypeScript/blob/master/src/server/protocol.ts#L18 by providing CompileOnSaveAffectedFileList and CompileOnSaveEmitFile commands.

I'm consuming thoses commands in https://github.com/angelozerr/typescript.java and it works well. It should be fantastic if typescript-lsp could provide a custom LSP command for this feature.