So when I cook the burger patties on a frying pan (I lack fancier methods of cooking them), the bottom tends to get burnt pretty black. Yet when I bite into the burger, you can see that it's only a thin layer of burntness outside, but the inside was medium-rare.
Is there a way to prevent them from being burnt too dark? I'm not sure if eating too much burnt stuff is good for the body. Perhaps it's a different oil that I must use? (I'm using canola oil atm).
I would try frying at a lower temperature. This will slow the burning of the exterior, but you'll need to re-assess how long to fry them to achieve medium-rare doneness.
You shouldn't need any oil when frying a hamburger, but you definitely need to lower the heat. I fry my hamburgers on Medium-Heat in a stainless skillet. Depending on the size of the patty*, I put the patty into a heated up pan and flip after about 6 minutes, then flip again after another 6 minutes, then again after 2 and then it should be done 2 minutes later. This will make a medium-well burger with a little exterior char, but still juicy and tasty.
*I usually do about a 1/3rd pound patty and I let the meat marinate in a generous amount of Worcestershire sauce before hand.
I've recently started making smashed burgers -- http://www.seriouseats.com/recipes/2014/03/ultra-smashed-cheeseburger-recipe-food-lab.html -- which you make thin, then double up. It makes it much easier to get well-done but not overdone burgers. You get a good taste because there's a lot of surface meat getting browned.
With thick steaks, the trick is to flip them once every 30 seconds, but burgers might fall apart if you try this too much.
Also, is the meat really cold out of the fridge? You may want to leave the meat out 10 to 15 mins to warm up a bit. Really cold meat will burn or toughen in a skillet.
To avoid the middle being undercooked, once I sear the outside of the rissole, I cover the hotplate in foil, just loosely - this creates a little oven that the rissoles can cook through - it also improves the time taken to cook them.
I recommend cooking the meat on top of an onion slice. You can cover the pan as well, though I don't. Use a clean onion slice each time you flip the meat over, then it'll never burn and the meat will be well done.
Its not the pan temperature, its the oil. You only need enough oil to cover the bottom of the pan for both sides. When the pan is hot you add the oil, after 30sec to a min add the patty. (Should be seasoned already). Let it cook for two minutes. Before the oil cooks all the way out but is 80 percent gone flip the patty and add no more than 1/8 a cup of water. This will keep the juice in the burger, keep the bottom from burning and is a great test. Once the water cooks out, flip repeat, and its done. I recommend adding seasoning to the water because the water will wash out the seasoning on the burger.
Combine @Onepotmeals answer (letting the temp of the meat rise before cooking) and @JacobG (lower cooking temp) and you have the canonical answer.
Also you said you nothing fancier than a frying pan--don't stress on that. My best meat cookery has been in my iron frying pan: stove top and in the oven.
