Glad to see this getting some love - it's middle notes, as you mentioned, are where it shines. And yes, over applied, it can get very gag-inducing...I've found that only 2 sprays applied as far away from my nose as possible is the best way to apply it.
As far as what notes I'm actually smelling when I wear it - well I've given up trying to name them. It just smells like medicine to me - which is not far off from what the perfumer/designer intended when they created it (to smell 'like a drug'). A fascinating, absurd, dark, surreal medicine.
If you like it, you should also try smelling White (which is basically CdG toned down, with more cinnamon up top and a tart pomegranate note) or the hard-to-find CdG Eau de Cologne (the juice is greenish), which is the same scent as CdG but sheer and refreshing...you can splash it on without fear of medicinal claustrophobia.