Carbon build-up around burners on a flat-top stove is a common problem. There are many products on the market that can help, but they all seem to leave something behind. A method that I recently stumbled upon is to use a razor blade to gently remove the carbon.
To clean up the mess, I pre-soaked the build-up with isopropyl alcohol, then scraped gently with a razor blade scraper. I kept the razor at a low angle and held steady pressure to prevent scratching the surface of the stove. Overall, I was very pleased with the results and it took much less work than I have previously put in with compounds and abrasive pads.