What do you recommend, cooking a burger on the stovetop or in the oven?

George Foreman grill, about $ 10-12 at Walmart, Target or even CVS

Stovetop. I have honed my burger craft over the last three years, and have experimented with every cooking type imaginable, from stove top to grill to smoker to deep fryer to even sous vide. I have found what works the best is the aforementioned cast iron skillet on top of the stove. Why? Because of the even heating, and heat retention that cast iron gives you. Another bonus is that the fat does not go anywhere like a grill. The grill grates cause the burger to lose so much juice down into the fire, whereas the skillet, all the fat pools around the burger and essentially shallow fries the crust. You will never get a burger as juicy as one cooked in a skillet, period. The broiler may seem like a good thing but it is not. The stove top will allow the burger more than enough heat and you can control the process a lot better than the broiler. As for he actual cooking process, well, that is another question entirely.

Definitely the stovetop. Agree that cast iron works well but I would recommend investing in a grill pan. Awesome for burgers, steaks, seafood, chicken, and veggies!

Definitely on the stove and I like to use my cast iron. Super hot pan, 4 minutes to get a great crust, flip, add the cheese, cover and after two minutes (again the crust) turn off the burner for three minutes more, rest. Perfectly done. My (11 year old) son prefers my cheeseburgers over any restaurant!

If you can't use a BBQ , then invest in either a cast iron pan with ridges or a grill with ridges that you can put over your stove top.
IF you want to use the oven I suggest broiling rather than roasting or baking.

An 80/20 mix of lean to fat is a great way to error proof your cooking. On the stovetop, in whatever your trusty frying pan may be will work great. Just remember two things- 1. Less handling, barely pressing the patty together while the meat is still very cold will help keep the fat from activating which makes your burger less chewy. A good coating of salt and pepper will add flavor, do both sides!
2. Lay your patty on the hot pan (350 degrees is perfect) and leave it alone! Once the protein cooks it will release from the pan and have a perfect sear. Flip and do it all over again. Don't press, smush, or squeeze. Enjoy that juicy burger!

Stovetop - preferably cast iron that can get relatively hot and get you a good crust on the outside. If you like our burger rare - medium give grinding a shot. You can get a nice hand grinder easily and mix Equal parts chuck and sirloin (or any mix of fatty and rich vs very meaty like flat iron) and 1 tsp kosher salt per 1 lb of meat. You won't have the worry of foot borne illnesses associated with ground beef. Good luck!