I guess I just don't understand the architecture of these things. Additionally, that project seems very poorly documented. I can't tell what many of the functions do :-/
One side-note: The way LWIP and the Espressif stack go together are great for things that want BSD-style sockets, but, they really do chow down on RAM usage. I guess I'm used to doing TCP on an AVR with 4k of ram. I really need to just accept that the ESP's got 40k and move on with my life.