Mac Pro 1,1 running any kind of latest version of macOS
It sounds like you have a mac pro 1,1 like mine with Quad core 2.66GHz processors.

It is correct about the 32 bit EFI on these systems but there is a workaround with a re-written bootloader for the MacPro 1,1 that will enable you to run El Capitan (last OS before Sierra) flawlessly on your Mac Pro. It even includes a script that will check and fix the bootloader each restart which means you can install system updates without having to repatch your system each time. 

You will need a better video card. I suggest one with at least 1 GB of video RAM. There are a lot of PCIe graphics cards out there that will work in a MAC but they don't boot EFI which essentially means you will get a black screen during boot up until the boot has completed - then the video card will switch on. This is only an issue is you have to boot into a separate hard drive or a Bootcamp partition. You won't be able to see your selections once booting and holding down the OPTION key right after the chime. You will have to find a fully Mac Compatible video card if you want boot screen graphics. 

To get El Capitan to install on a MacPro 1,1 you need several things: 


an 8GB USB
a copy of the Mac App Store El Capitan installer
at least 12GB of RAM
a video card capable of running the latest OpenGL and with at least 512MB of video memory - 1 GB is better. 
You will need to rebuild the installer to work with a MacPro 1,1 and to use the 32bit EFI translator. Instructions and the links for the downloads can be found here. 


No Mac OS after Lion will install on a MAC PRO 1,1 without editing the original install file. Apple places checks in the installer so that it will only work with certain motherboards. Part of the tweak to the installer involves adding your system's board ID to the installer so it will pass a check and install. There are tools and instructions for how to rebuild the installer for Mountain Lion (uses MlPostFactor which is a "patched" install and not native - certain kernel level functions will fail such as "Bootcamp" also Little Snitch and Hands Off will not work) Mavericks, Yosemite and El Capitan. 

El Capitan is about the easiest OS to modify because Captain Pike's Script automates the entire process. No real advanced knowledge or manual editing of system files is required. I followed the instructions, downloaded the script, let it do its thing and I have a flawless MacPro 1,1 running OS X El Capitan. 

I also recommend adding an Airport Extreme wireless card and a Bluetooth card for your machine. I don't think you will find a Bluetooth card for the MacPro 1,1 that will work with OS X's "Handoff" feature but if you are using an iPhone you will want it to be able to sync with your Mac Pro without having to plug it in, so you will need the AirPort and the Bluetooth to get the most out of your Mac. They are easy to find, relatively easy to install and are fairly cheap. Instructions for how to perform these upgrades can be found here.  

Incidentally El Capitan is the last OS so far that you can tweak to run on an old MacPro 1,1. If you are having trouble finding a Mac App Store copy of El Capitan, Google is your friend for finding torrents. Just make sure to include the words "pirate" and "MAS El Capitan torrent" in your search terms.