1/8 inch above lip of gutter is correct. This allows for water to exit the pavement and enter the flow line of the gutter.
If your compaction tests for the native soil and aggregate sections meet the specified standard, i.e., 95% relative compaction, then the 1" above the concrete lip is to high. Compaction is irrelative if this was an overlay of existing asphalt.
This creates an obvious hazard to "catching your toe" and causing a fall. Liability can be an issue here.
The solution is to lower the edge of asphalt to meet the 1/8 requirement. Again, 1/8 inch is optimal, with up to 1/2 inch maybe acceptable to allow for settling of the A/C over time.
There are propane indirect heaters that can accomplish the re-heating of the A/C in localized areas, thus making the A/C pliable again and then it's just a matter of workers raking away the excess and removing it and then re-compacting the A/C to seal it again.
The contractor probably did not figure removing asphalt into his bid, however, it seems the contractor gave you a "honest section" with the 1 inch above lip. Coring is also a quality control technique that will verify that specified standards have been met by the contractor.