The best way to cook burgers on the stove differs from the way on a grill.
First off, you might want to make the patties thinner and wider. You might also try making two half size patties and going for double burgers because the extra caramelized and browned surfaces are so packed with flavour.
Whatever you decide, the secret to good griddled burgers is to use a heavy cast iron or a steel iron pan, and to press the burger down after the first flip, and then again lightly after flipping again. Turn a few times, the old advice to only turn meat once has been definitively proven wrong.
Use medium to medium/high heat stead of high to get a final result where the inside is cooked to your preference without overcooking the outside - this seems to be your problem, simply too high heat.
Covering the pan is good for melting the cheese on a cheeseburger, though, if doing so make sure to put the cheese on about 2 minutes before the burger is fully cooked or you'll end up with the insides overdone by the time the cheese is melted.
On a barbecue you avoid pressing the meat, of course, and the cheese will melt fine uncovered. I tend to cover a Weber style grill for the last minute to melt the cheese, just before toasting the buns on the grill.
This is a great site for everything involving barbecuing and even grilling and roasting meats, it's right up on the current science, and he tests everything out in practice.
amazingribs.com