Mix yogurt, relish, sugar, vinegar, ketchup, and mustard in a bowl until smooth. Chill in the refrigerator while cooking so flavors blend nicely. Cold sauce tastes better when served over hot burgers.
Sprinkle salt and pepper on both sides of the patties. Let them rest for a few minutes at room temperature so they cook evenly. Avoid overworking the meat to keep patties tender.
Heat olive oil in a skillet over medium heat and cook onion slices until golden on both sides. Add cheese slices to two onion rounds and cover briefly so cheese melts gently.
Cook patties in the same skillet until browned and cooked through. Flip once and place cheese slices on two patties, letting them melt naturally from residual heat.
Place an onion slice on a plate as the base, add a plain patty, then a cheesed patty, and top with the onion slice holding melted cheese. The onion replaces the traditional bun.
Drizzle the chilled sauce over the burger or serve it on the side for dipping. Serve immediately while everything is hot and melty.