Roast the garlic for the compound butter. Preheat the oven to 375°F. Cut the top 1/4 off the head of garlic, exposing the cloves. Drizzle a bit of olive oil or avocado oil on top of the exposed garlic cloves. Wrap the garlic in aluminum foil and bake in the oven for about 45 minutes. Then, remove the foil pouch from the oven and allow the garlic to cool in the foil until it’s cool enough to handle.
Make the compound butter. When the head of garlic has cooled, add the roasted garlic to a bowl with the butter, Worcestershire sauce, and prepared horseradish. Mix together well. Wrap the butter mixture tightly in a piece of plastic wrap, forming it into a log shape. Freeze the butter for at least 2 hours.
Make the butter burger balls. Add the ground beef to a large mixing bowl. Grate the frozen butter directly into the bowl with the beef. Mix the butter and the beef gently so as not to melt the butter. Then, roll the beef mixture into balls that are about 3 to 3.5 ounces each, or slightly larger than the size of an egg. Refrigerate the butter burger balls until griddle or skillet is preheated.
Heat griddle or skillet on medium heat. For griddle cooking, the griddle should be about 400-425°F.
Cook the smashed butter burgers on the first side. Place the ground beef balls on the preheated flat top grill or in the skillet. Top each beef ball with a handful of sliced onions (if using). Use a stiff spatula or a burger press to smash the ground beef balls down on the cooking surface, until about 1/4 – 1/3" thick. Sprinkle the top with salt and pepper. Allow the burgers to cook untouched until the outside edges of the sliced onions are lightly browned, about 3-4 minutes.
Flip and continue to cook the butter burgers until done. Flip the burgers to the second side, so that the sliced onions are now in contact with the cooking surface. Immediately after flipping, top each burger patty with a slice of American cheese (if using). Cook the butter burgers an additional 2-3 minutes, or until cooked through.
Butter and toast the buns. Spread about ½ tablespoon of butter on the cut side of the hamburger buns. While burgers are cooking, place the buttered buns on the griddle to toast. When buns are done toasting, remove them from the griddle.
Serve your butter burgers. When burgers are done cooking, stack two burger patties on top of each other to make a double butter burger. *TIP - As you're removing the double burgers from the griddle, tilt the spatula a bit to allow some of the excess grease to drain off before adding the burgers to the bun. Place the burgers on the bottom bun, and top with your favorite burger toppings and the top bun. Serve immediately.