I decided to try asking Aaron directly and turns out he's awesome about answering questions. Here's his answer:
"Appreciate the kind words. To my knowledge Andy used an EVH 5150III for all of his reamping, (rhythms and leads). Single SM57 on a Mesa 4x12 cab with the standard Vintage 30 speakers. I did a little EQing in mixing, mostly notching out a couple odd resonances and some very gentle tone shaping, but the main component of clarity is definitely a combination of the playing itself, and the fact that Andy loves midrange in his guitar tones. The high mids are where note clarity lives! Try boosting the mids on an amp higher than you think you like, but then compensate afterwards in your DAW by reducing the boxy low mids, and see how you feel about it. I've also had success actually cutting low mids on a DI before reamping, which then lets me boost the mids on an amp without bringing up the mud."