Serve Anonymous Pages when users are logged in for some pages
You may implement ESI block caching. 

Have a look at ESI module  and varnish basic tutorial. You would be able then to cache whole pages except the blocks with session data.

Additionally, @Kevin comment points a complementary way: using Redis or Memcache will substantially speed the pages when logged, by caching data and objects in server memory to reduce the number of times the data source must be read.