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:
The best way to understand Friedman's and 1950s Chicago methodology is to consider it more an operation research outfit than an economics department. The group consolidated in the war-time SRG, an OR group studying the mathematics and statistics of weapon ordnance (check out Machine Dreams for a good history of it).
As with all OR groups, the Chicago group put the priority on usefulness. What matters in economics to Friedman is that the models predict well. Even if the premise of the model sounds wrong, it's fine to act "as if" it was true as long as it predicts data well.
This went against most of past (and future) economics. The predominant philosophy of science in economics hasn't changed since John Stuart Mill. In short the idea is that data and experience is misleading, that the job of the economist is to isolate causal mechanisms whose effect may be masked by countervailing forces when we look at data alone.
This results in a science that spends more time thinking about first principles (assuming everyone is rational...) than data itself.
Maki makes a good case for it. There is a nice blog-post about it on "a fine theorem".
The underlying belief among most "Chicago" economists is that "markets work," and most of public policy should mostly be directed toward improving their efficiency, e.g., by a "steady" monetary policy. Examples of economists include Milton Friedman (Capitalism and Freedom), and Ronald Coase (who postulated that most legal results are dictated by economics).
The opposite, "saltwater" school of thought (on the east and west coasts) recognize that markets often fail, and therefore focus on using government solutions to compensate for such failures (e.g. using fiscal policy to get the world out of the Great Depression).