I love medium rare hamburgers, but the USDA says no, cook until done, which the USDA says is 160 dF.
Well, not only is all the flavor gone at 160 dF, but I'm afraid I'll break a tooth!
The USDA recommends this since most people cook their hamburger at a high temperature, which means it's on the grill for just a short amount of time before we eat them.
Therein lies the problem - high cooking temperature means a short amount of time to pasteurize the meat so it is safe.
So we can cook at a high temperature until the meat has reached a high temperature, and the meat will be pasteurized.
Or we could change the cooking formula, and cook at a lower temperature for for an increased length of time, and still pasteurize the meat. It's our choice.
We don't have to blindly cook our meats to high temperatures because we've got a Big Green Egg that allows us to precisely control the temperature that we use to cook the meat at, and that means we can cook it at a lower temperature, but for a longer period of time, in order to pasteurize the meat.
That means I can cook my hamburgers for a longer time, but at a lower temperature, while still pasteurizing the meat.
It means I can have a medium rare hamburger and be safe.
I do a lot of Sous Vide cooking, so I am familiar with this approach. I don't know what temperatures to recommend to you to cook your meat in an oven, which is what the Big Green Egg is when the lid is closed, in order for you to have safe meat.
I do know that I can cook it in a pouch, in a waterbath with a precisely controlled temperature of 131 dF, cook it for 4 hours, and have pasteurized medium-rare meat. I can then finish it on the Big Green Egg to toast the exterior (Maillard Browning), and I've got a wonderful, safe, tasty hamburger.
More Hamburger Insight Here
Now, if you want to do the whole thing on the Big Green Egg, then you need to insert a temperature sensor (I would recommend a thermocouple device because it can take the surrounding cooking chamber temperatures OK too) into the ground beef. Start your clock at the time it reaches 135 dF, then cook for four (4) more hours at this temperature and you should be good to go. Time charts vs temperature for pasteurization are available at USDA web site.