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Yes, there is something you can do. I guess it's not really a trick... just not well known. There are products on the market designed to seal scratches and imperfections, and to restore the smoothness to plastic laminate. They are wax products, similar to automobile polish. By sealing the surface, staining is reduced and cleanup is much easier. Gel Gloss is one such product. I am sure there are others.
Automobile wax would also work as a sealant, too. However, there may be additives in auto wax that would make it undesirable to use on food contact surfaces.
As with all wax products, it will wear off over time and need to be reapplied.
Sorry about your countertop! I hope you and your child weren't too traumatized by this accident. Fires are always scary!
There are a number of possible solutions to your problem:
(1) Purchase a wooden or plastic cutting board and just leave it on the countertop to cover the damage! There are some cutting boards that look so good you would want to leave them out, even if your countertop was undamaged. One suggestion is the "John Boos" maple cutting board. It is unique because it has a small "backsplash" as well as an overlap on the front edge of the counter. It is reversible so you will get many years of use from it. These quality cutting boards are less expensive than a new countertop... and don't require any carpentry skills to install!
(2) If you do a little hunting around you might be able to find a
pre-made inlay. (See graphic below). I didn't have any luck
finding one online, but it's possible a home store or kitchen
remodeling/cabinet showroom would have this product. If you
find one, follow the instructions in (3) to cut the countertop...
just ignore the cutting board prep.
(3) You could take a thick piece of butcherblock and build it into your existing countertop. Though I admit to never having done this, it should be rather easy to accomplish. The first thing would be to find a suitably-sized cutting board. A piece of butcherblock would be ideal provided it is at least 1 1/2" thick. First, prepare the cutting board by cutting a notch around the perimeter of the bottom about 3/4" wide by 3/4" deep. This notch is called an edge rabbet. This is most easily and neatly done with a router, though this can also be done with a table saw, making two cuts... one about 3/4" deep on the edge and about 3/4" in from the bottom. Then, make a second cut all the way around the bottom to meet the first cut, leaving a 3/4" x 3/4" notch all around the bottom.
Now, cut a hole in the countertop, all the way through, slightly larger than the bottom of the cutting board. (This will be about 1 1/2" smaller in each dimension than the original size if you made your cuts according to my suggestion above.) Once done, the cutting board will set nicely into the countertop, covering the hole with 3/4" of overlap around the edges. This plan has the cutting board raised above the countertop about 3/4" if it was originally 1 1/2" thick. If you would like it raised a little less, make the notch deeper.
One problem with this design is that any spills on the countertop will leak around the cutting board and into the cabinet below. If you think this might be a problem, set the cutting board into the countertop with either latex caulk or plumber's putty. Plumbers putty will stop leaks, keep the cutting board a little more steady but allow you to remove it for cleaning when necessary... though you should be able to perform most cleaning tasks without removing it. If you would like it set more permanently, use latex caulk. It will still be removable, but you will need to slide a putty knife carefully under the edge to free the caulk's glue-like bond.
On possible drawback is the cut-out section of countertop might be tend to sag over time if there is no nearby support underneath. To remedy this, glue-and-screw 4"-wide strips of 3/4" plywood under the counter around the perimeter of the hole. They will firm it up to nearly original strength.
We went to Home Depot and they had a frosted glass inlay cutting board. We bought it and my husband came home and started cutting (sawing rather) the old countertop out and with-in an hour I had a nice looking countertop again. It was very simple, he said , and almost anyone could do it. SO, instead of paying over $120.00 for a new countertop because there we some burn marks, we paid 15.00 and now have something that is nice looking and it is right next to the stove for cutting convenience. He did the cutting as you suggested except this inlay came with the metal ring, ready to install.
I wanted to send you a picture of our countertop after my husband put the inlay in it....just in case anyone else ever has a problem like ours you can give them this idea.
Thanks again for your help!