There is really no way to repair a bad burn in a plastic laminate countertop, but you can still save the counter. In this segment I used a plunging router to cut out a very flat hole in the counter, just the right size to put a cutting board right where the burn was. Don't forget to carefully seal the entire cut-out with silicone caulking to ensure that no water will get under the good part of the counter and swell up the plastic laminate.
I had the pleasure of using the small Freud 1/4-in router, model FT 1000E. This thing is only slightly larger than the standard rotary tools that are common on the market, but it runs at 7 amps. That's a rotary tool on steroids. In fact I was cutting with a 3/4 inch straight plunging bit and it didn't even put drag on the motor. The motor can be used freehand, or with a series of useful bases, including the plunging router base. Quite a tool!
One caution: start in the middle of the work area and work out to the limiting frame. That way you will always have at least half the router base on the counter top. If you start on the outside and work towards the centre, you will fall into your own hole, like cutting off the tree limb that you are sitting on.