If you need to read the entire content of the P-Asserted-Identity header of an incoming INVITE, be aware that you should change the sofia profile by adding a param like:
param name="p-asserted-id-parse" value="verbatim"
FreeSWITCH will populate the variable accordingly and make it available with commands like (e.g. with lua):
PAID = session:getVariable("sip_P-Asserted-Identity")
If you don't add this parameter, you'll get the default behaviour, which is just filling the variable with the P-Asserted-Identity URI username part.
Possible value are: "default", "user-only", "user-domain", "verbatim", which I think are self-explanatory.
A recent reference here.