Combine the ground beef with the olive oil, salt, and plenty of black pepper.
Divide the meat into two 9-ounce patties and refrigerate until the grill is ready. (Don’t do this more than an hour in advance.)
Combine mayonnaise, horseradish, and lemon juice in a mixing bowl and season with black pepper. (You can do this ahead of time and store it in the fridge.)
Heat the grill to high.
Take the hamburgers out of the refrigerator 5 to 7 minutes before you are ready to grill them.
Place the burgers on the preheated grill and cook 1½ minutes (for rare).
Give the burgers a quarter-turn to “mark” them, and cook 1½ more minutes.
Flip the burgers over and cook 1½ minutes.
Rotate a quarter-turn to “mark” and cook 1½ more minutes.
Transfer the burgers to the grill’s top shelf or to a cooler section of the grill and cover each one with a slice of cheese.
Turn the grill off and shut the lid.
After 4 minutes, open the lid. The cheese will be melted and the burgers cooked rare to medium rare. Toast the buns, if desired, and place a burger on each.
Spread plenty of the horseradish sauce on each burger; it should drip down the sides.
Top with Crispy Onions and season with freshly ground black pepper.
Slather more sauce on the other half of the bun and place it on top of the burger.
Grab a cold beer or iced tea and get ready to make a mess. This is not a dainty meal.
For the Crispy Onions:
Place onion rings and oil in a small sauce pot. (Don’t worry if the rings break apart; it won’t ruin the final product.)
Bring to a boil over high heat, and then reduce heat to a very low simmer. The heat releases the onion’s natural sugars, and in essence, causes them to melt.
Turn the onions with a fork every thirty seconds or so, and cook until they turn golden brown, 12 to 15 minutes. Adjust the heat if needed to maintain a low simmer.
Remove the onions from the oil and arrange in a single layer on paper towels. (At this point the onions won’t yet be crispy, but I promise you that after a few minutes, as the caramelized sugars cool and harden, the onions will become deliciously crisp.)