So question is: is there a way to make a client-server application on haxe, without branching code for different platforms?
Best How To :
What you are looking for is Haxe Remoting, which is part of the Standard Library
Haxe remoting is a way to communicate between different platforms. With Haxe remoting, applications can transmit data transparently, send data and call methods between server and client side.
As side note, if you are looking into creating a actual website with client/server in one project you might want to look into http://www.ufront.net