Prepare the special herb sauce by whisking together the mayonnaise, sweet pickle relish, ketchup, yellow mustard, sugar, onion powder, and paprika in a small bowl until smooth. Fold in the finely chopped fresh chives. Cover and refrigerate until needed.

Cut the onion into thick (about 1/2-inch) even slices. You want central, sturdy rings to serve as buns. Save any remaining pieces for another use.

Heat a large cast iron skillet or griddle over medium heat. Add the oil. Once hot, carefully place four of the thick onion rings into the skillet. Cook slowly for 6-8 minutes, allowing them to deeply caramelize and char on the bottom without burning. Carefully flip the rings with a wide spatula and season lightly with salt and pepper. Cook for another 5-7 minutes until deeply caramelized and tender. Remove and set aside warm.

Divide the ground beef into two equal portions. Gently form each into a patty that is slightly wider than your onion rings (they shrink). Create an indent in the center. Season generously on both sides with salt, pepper, and garlic powder (if using). Wipe out the skillet.

Return the skillet to medium-high heat. Place the seasoned patties into the hot pan and cook for 3-4 minutes until a deep brown crust forms. Do not move them.

Flip the patties. Immediately top each patty with two slices of American cheese, overlapping them to cover the surface. Reduce heat to low and continue cooking for 2-3 minutes until the second side has a crust and the cheese is completely molten and draped.

To assemble, place one caramelized onion ring bun (the larger ones work best for the base) onto each serving plate or board. Spread a dollop of special herb sauce over the onion. Carefully lift a cheese-draped beef patty from the skillet and place it on the sauce-covered onion bun.

Optionally spread more sauce over the top of the melted cheese, then top each burger with a second, smaller caramelized onion ring bun. Press down gently and serve immediately with plenty of napkins.