What's unique about the Chicago School's Methodology? What is the methodology of the Chicago School and what makes it unique among economic schools of thought?
I would submit that the premise is flawed. There is no methodological criteria that unified the Chicago school and separates it from the rest of economics. I think this is true even for Friedman, whose methodological paper you refer to. If you look at his macroeconomic work such as his Monetary History of the United States, Theory of the Consumption Function or his famous presidential address to the AEA,there is nothing in it which is different from the methodology of mainstream economics. I think the same can be said about Stigler's work on industrial organization.

The Chicago School was united by its institutional links and its faith in capitalism and not by a distinctive method. 

And it would be wrong to think of the School as a homogenous entity. See for example this article by Patinkin on the diversity of views on the quantity theory of money:

http://stevereads.com/papers_to_read/the_chicago_tradition_the_quantity_theory_and_friedman.pdf