Okay - if there's anyone who comes across this issue in the future, here's what I did to fix the issue.

First, what I have pictured is a stem extension for a Delta faucet. Removing this item required me to turn the water off to my house and disassemble the shower escutcheon (metal sleeve that this plastic bit fits in to). Once I had it disassembled it came out of the actual valve in the wall.

From there I went to home depot and purchased the following:

Stem Extension

Delta Stem Assembly

Gasket kit for stem.

Since I already had the water off I went ahead and replaced both sides of the shower handles.

I disassembled the valve, took the old parts out, and put the new stuff in in the same way it came out. The issue at hand was someone took the old shower handles off of the stem extension and didn't replace the long metal bolt that screws in to the brass section of the stem assembly. They then used a "clamp" style shower handle that broke the stem extension.

All and all, including a new shower head, I was out approximately $170. That includes new shower head, shower head extension, and two new ceramic handles. Total actual fix for these parts (assuming you don't have to replace shower handles) would likely be more around $60.