Thanks for the help, Bob!
A big reason why I try new hardware like this is just for learning, so that I can do more advanced things in the future. So even if there's no noticeable speed improvement by switching from EEPROM to FRAM, I'll still benefit from the project because I'm learning about FRAM, more about SPI, and also I'll be able to solder my first soic.
For my hardware, I have the webserver running on an Arduino Due (SAM32 80Mhz), with a standard Arduino ethernet shield on top. I'm not knowledgeable enough to make my own TCP/IP stuff yet.
It mostly suits my needs, but I'm always keeping speed in mind because the web page is complex, since it has to check for so many different variables on each load. At first I was using one of the cheaper Arduinos that had an (I think) 16Mhz processor, but after a while the webpage became sluggish on that due to the complexity. But when I switched from that to the Due board, there was a very noticeable and pleasant speed improvement. The page currently takes about 900 ms to load, which is fine for now, but I wouldn't want it any slower than that, and I'm always adding new features.