How to avoid burned layer on seared hamburger?
Make sure you flip it reasonably frequently; that'll actually make it come up to temperature faster inside, so the outside won't have as long to burn.

Also make sure you haven't made your patties excessively thick. 10-15 minutes on medium sounds like a fairly long time for normal sized burgers, so possibly yours are on the thick side. If you do really want thick patties, you're probably better off using a more gentle cooking method like the oven for some of the time. You can bake until nearly done, then sear to finish, or start with a sear then bake til done.

Beyond that, if it's burning on the outside before it's done on the inside, the normal fix is to simply reduce the heat, so you cook more slowly, giving the heat time to reach the center without burning the outside. But at 10-15 minutes already, maybe you don't want to go much slower!