Two websites, same url, same cookie

We are thinking about rewriting a web application. The old application is written in Java and the new one in c#. I want to rewrite it incrementally. So let's say this web application has 10 screens:

First rewrite screen 1, deploy it. Use only 9 screens from the old web application, and 1 screen from the new web application. NExt rewrite screen 2, deploy it. Use only 8 screens from the old web application, and 2 screens from the new web application.

For this to work both applications probably must run on the same url and reuse the cookie.

IS this even possible?