I grew up five blocks from the Johnnies Italian beef joint (it's located in Elmwood Park, a suburb of Chicago). I have spent the past six years trying to recreate their recipe, so far without much success. I have even done mail order from some of their competitors (Johnnies doesn't ship), in order to refresh my memory of what it should taste like (although their competitors' beef is not quite the same).
If anyone has a complete, reasonably authentic, Chicago-style Italian Beef recipe, I would pay good money for it.
BTW, here is a short video, showing Johnnies. I found it really interesting because of the brief glimpse into the kitchen where they actually make the beef. I had read all sorts of preparation tips, like roasting it, and then recombining that with the juice. Well, as you'll see if you go to the 2:10 mark of this video, that is not how they do it. Therefore, sous vide, or even a slow cooker, should do a pretty good job of replicating their technique (I like the idea of sous vide because it should do a better job of concentrating the oregano and garlic flavors).
Someone who knows their cuts of beef should be able to tell us what cut we are looking at.
Here's the closest I have been able to come, but it still isn't quite right.
"Johnnies" Italian Beef
3½ pound beef roast (the OP's choice of Eye of Round is probably even better)
½ peeled bulb garlic (either cut up and put in roast, or puree)
¼ cup extra virgin olive oil
1 teaspoon Italian seasoning
1 tablespoon oregano (or more)
1 teaspoon black pepper
2 teaspoons sweet paprika
2 teaspoons crushed red pepper
½ teaspoon sea salt
Broth
¼-½ jar (16 oz) Pepperoncini peppers, with juice
24 ounces beef stock
I have tried it using garlic powder, but didn't like the result. I've tried various beef stocks, but am quite certain I don't yet have the right product. I have mashed the garlic and rubbed it all over, and I have simply studded the roast with garlic. I have used more oregano (I don't think it is possible to use too much).
I am getting a little closer, but the garlic and beef stock are the two main things that I don't think are quite right.