Don't know how authentic, because my only experiences in Chicago involve bad experiences at O'Hare, but Cal's Chicago Style. I've not tried their dogs but I do enjoy the Italian Beef.

I didn't know the California was supposed to have fries so you probably won't like Taqueria Rafita's, since they use diced, but I enjoy their version.

Is the Calfornia different from the Mission Burrito? I risked my life to get a Mission Burrito in the Mission District of San Francisco and I don't recall fries on my Mission Burrito.

And while we have plenty of BBQ around here.... Edge Craft is Central Texas style and has even been given praise by, the Texas Monthly BBQ Editor Snob, Daniel Vaughn