When I was growing up, my mom always made us cheeseburgers with cheese stuffed inside of the burger rather than slicing it on top. It wasn't until much later that I discovered this wasn't just a Chevriere family thing but an actual style of cheeseburger, called a Juicy Lucy. Regardless of what you call it, it's delicious. (And I would also recommend experimenting with other types of cheese besides Cheddar, like blue cheese or taleggio.)
In a small skillet over medium-low heat, combine the onions, olive oil, red wine, and pinch of salt. Cook for about 45 minutes to caramelize, stirring constantly.
While the onions are cooking, you can prepare the burgers. In a large mixing bowl, combine the ground beef, horseradish, red wine vinegar, and salt and pepper. Mix well to make sure all the seasonings are fully integrated with the meat. To form the patties, take about a fist-size ball of the ground beef and in the middle create a divet. Repeat for the other two patties, reserving a small amount for each to cover the hole. Fill the divet in each of the patties with the grated cheese, then cover with a thin layer of meat, making sure the sides are well sealed
Cook the patties in a cast iron skillet or medium-sized pan over medium-high heat for about 5 minutes on each side, or until they reach desired doneness. To compose, place the patties on the buns and top with some of the caramelized onions.