Actually, it's a lot worse.
The first shop I took it to thought it was the starter engaging. Since they don't do electrical work, they sent it to another shop.
I got a call this morning that it's not the starter, it's the flywheel. I had my rear knock sensor replaced a while ago. Apparently, the old knock sensor fell into the housing and eventually tore up the flywheel.
I called the shop that replaced the knock sensor, but the manager isn't there today. I'll talk to him tomorrow, and if I don't like the answer they give me I'll probably seek legal advice.