There are many styles of burger, so there is no single answer. Steamed sliders are going to have different constraints than a smashed burger compared to a pub style burger.
For burgers that are griddled (for the purposes of this post, cooked on a hot, flat surface), the main issue is how much heat can get into the burger, and how fast; secondarily, there are issues of sticking and general maintainability.
Some hamburger cooking methods rely on sustainable, very high heat to sear the outside of the burger, and give a good crust, even while maintaining a less cooked intererior. This is dependent mostly on the total thermal mass of the pan.
Stone - Lots of mass, but low thermal mass. Doesn't transmit heat very fast. Also requires considerable care and maintenance to prevent sticking.
Pan - There are so many types, it is hard to give a single characterization. Thicker, heavier metal tends to be better. See Navigating the different types of cookware - new kitchen advice for some general advice.
Cast Iron - Generally has high weight, and very good thermal conductivity (compared to any non-metal), so often ideal for high temperature searing of hamburgers. Requires seasoning and maintenance.
Charcoal (or propane) grill -- can develop extremely hot radiant heat, which is ideal for some forms of burger cookery, but cannot be done inside.
However, there are so many different types of tools and styles of burger, the quality of outcome is going to depend on your skill, what you are trying to achieve, and how you use the equipment you have as much as the specific pan that you use.
Similarly, the flavor is going to depend, not only on how you cook (which can develop the seared, brown, meaty flavors), but how you season the meat, and the quality of meat you start with. This is again as important as the particularly pan that you might use.
You're right with your assumption that surface bacteria should to be killed and by the process of grinding meat the surfaces bateria mixes into the interior. BUT:
There is also steak tartare which is basically minced beef consumed raw. The trick is to get the meat from a trustworthy source, store it at low temperatures, process it in an environment which is as clean as possible und to do it fast. It's not perfectly safe like eating completely cooked meat. You have to weight up the risk of getting food-poisoned and the better food. A compromise suggested by John Dyer in terms of making steak tartare:
If you are really nervous, a trick I have heard of is to start with a
really thick piece of beef. Then sear it on both sides in a hot pan.
At this point the outside would be deemed safe and the interior is
typically safe so you cut away the cooked parts. Then proceed to make
the steak tartare with the still raw inside part. As a bonus those
nice browned parts from the outside are a treat for the chef.
Since you are searing the whole thing anyway and the taste of the products after the Maillard-reaction is desirable you can sear the beef first, then mince the meat including the seared parts and then proceed as usual. No need to cut the brown parts.
For further reading: How safe is steak tartare?
Best Answer
This scum is made from proteins. Meat contains muscle fibers (the proteins actin and myosin) as well as some loose proteins swimming in the fluids within the meat (the cell plasma). When you cook meat, the protein-rich fluids are expelled (that's why overcooking makes meat dry). Under hot temperature, the proteins in the fluid coagulate, making it firm. It is especially noticeable in your burger, because there is more liquid flowing out quickly from your cut-up meat, but it also happens with whole pieces of meat, albeit more slowly. It is also very noticeable when cooking stock, because the stock is cooked for a long time and the liquid has time to come out from the meat.
The coagulated liquid can form a single piece (as it does with steak on a slightly oiled pan), but when it flows into water or oil, it mixes with it without dissolving, creating small loose particles. They float to the top, creating the frothy scum you describe.
The process is perfectly natural, you're not doing anything wrong. You can in fact eat the scum without any ill effects, but the taste isn't that great. So it is better to fish it out from the oil (or water when making stock). In dry cooking (aforementioned steak) it is exactly this stuff (after getting nicely browned on the hot pan) which makes gravy taste so well, together with the dripped fat.