An obvious sugestion is Katz's Deli, as Manny's be damned, there is nothing quite like it in Chicago. Duke the pastrami slicer for better effect.
One thing that I love in NYC that is pretty much non-existent in Chicago is Jewish/Dairy places. The problem is, they are getting to be pretty non-existent in NYC too. B&H in the gentrifying lower east side has its flaws, including none so sparkling sanitation, but it still fries up a mean blintz. Actually, I'd love to know myself where a good dairy place is these days in NYC.
In a similar vein, nothing in Chicago comes close to Russ and Daughters for noshing.
I cannot remember where I mentioned this, but breakfast/brunch at Balthazar is one of the great treats of NYC. You get to be "in" without the hassles of dinner. The pastries are outstanding and the room is perfect in its hyper-perfect too good to be Paris way.
For pizza, have you considered Otto?