Ya, I’m an out-of-towner (Seattle-ite, kinda) but I’ve eaten b’fast there a couple of times and outside the place on a light pole or something there’s a sign that says “Route 66 starts here” so I dunno ...
There *is* a restaurant next to the sign on Adams St. that marks the un-official start of Route 66 but that's not Lou Mitchell's and I'd have to walk by there on Monday to see if they serve breakfast. I'm drawing a complete blank on the name of the restaurant at the moment and I've walked by it dozens of times over the years.