I currently have an assignment to critically analyze Mohammed & Charlemagne and the Pirenne thesis. (note: I do not want academic criticism of this particular work explained).
I'm feeling somewhat lost because I'm used to reading respectable secondary sources and learning from them, not necessarily trying to pick them apart. It feels wrong to approach a book that an imminent historian spent months or years researching and composing, looking for flaws. The arguments seem presented in a reasonable way, and without knowing his evidence thoroughly it's difficult to see where it can be criticized.
I have a hunch my only option is to really line up his premises and the arguments he makes and look at those, but I'm feeling kind of stupid and inadequate :(
It may be helpful to not think of "critical reading" as "reading to criticize" but rather as "reading to understand a secondary source as a project rather than a repository of facts and ideas to be taken for granted". With that in mind:
First of all, LOOK AT THE BIBLIOGRAPHY. What secondary literature is he citing? What primary literature is he citing? People can come to radically different conclusions looking at any topic depending on what categories of sources they use-a history of Christianity under Zoroastrian rule that focuses on hagiographies is going to look different from a history that focuses on the archeological record is going to look different from a history that focuses on administrative texts(such as they have survived). What sources in Pirenne using, and how might they shape his thesis? Ask if his thesis would be different if someone told him to use a category of sources he doesn't use.
What are the assumptions he comes into the work making? Every academic work starts out from certain assumptions and if you look carefully you can usually find them. Even if it's just about periodization(and how he defines a period might be something assumed or not), it's possible to suss out what they are going for and how that might shape the way Pirenne is examining his sources.
What kind of intellectual framework is he using? To some extent question 1 will help you answer this, but it may be useful to look at the kind of framework he's using to analyze the data, whether you try to fit it into some recognized school of historical research(marxian approaches, annales school, etc) or just try to formuate a general sense of what intellectual tools he uses to analyze his source base. Does he focus on trying to reconstruct something historically? Trying to analyse things in terms of "the way people at the time thought"? Try to focus on economic understandings of history? Does he position politics as a major driver or as something conditioned by other forces? And how does that shape his conclusions about what is going on in his study?
How does he structure his arguments? You mentioned this, and it's a very good thing to do as part of critical reading to just break his arguments into tiny little pieces and see how they're put together. Sometimes you can even learn from a historian's writing style-maybe he really likes narrative discussions, or he just loves to dump piles and piles of data undisguised.
Now that you've done all that, you can put Pirenne back together again and ask a few broader questions. What makes him tick as a historian?(structure and assumptions help here, as is "why he is writing this") What kind of project is Pirenne undertaking? How is he framing and approaching the problems?(the issue of source base, framework and methodology are good to look at here).
Hope all that helps!
Having just written a primary source analysis as part of my coursework for a history unit at university, this helped me critically analyse a primary source. (link to 'How to read a primary source' on the side bar) http://www.bowdoin.edu/writing-guides/