What the heck is "Chicago Beef n' Eggs"? Unless I've been in the dark for the last 53 years, there is no such indigenous breakfast (or any other meal for that matter) dish by that name. Whoever sold you that bill of goods was pulling your leg.
As for your breakfast needs, the entire Roadhouse Roadfood clan (Buddy, Mrs., abe_froeman, Mr. Maki) heartily recommend L&L Snack Shop in Des Plaines (about 6-8 miles from Palatine, down Northwest Highway; Route 14). Excellent pancakes, omelets, corned beef hash and other breakfast related items. Everything is served with a couple of hunks of ham fresh off the bone, whether you ask for it or not. This is a classic Mom & Pop diner with daughters involved as well.
As for the Brat Stop, no offense to ALLGOOD, but most native Cheeseheads agree you can do better. Honestly, I've been to the Brat Stop many times and like it just fine. Many of my Contingent
colleagues have their own, lesser known favorites, which I will let them recommend should you be interested.
If you need food between Billy Mitchell Field and Palatine, there's plenty to be had on both sides of the border, so just ask.
On the other hand, given your short stay, I think we've covered every meal you can possibly eat on this trip.
Buddy L&L Snack Shop
456 East Northwest Highway
Des Plaines, IL