Prepare the burger sauce by mixing all the sauce ingredients in a small bowl. Cover and refrigerate until needed.

For the smash burgers, divide the beef into 8 portions (3 ounces each) and shape them into loose balls on a plate. Cover and chill while you prepare the grill. Cut a 3-inch square of parchment paper and set it aside. Preheat the grill to high (450°F to 500°F) and place a large cast-iron skillet or griddle on the grill to heat up. Alternatively, you can heat the skillet on the stove over high heat.

Drizzle the skillet with oil. If needed, work in batches to arrange onion slices into 8 small mounds (about 1/4 cup each) in the skillet, keeping them at least 2 inches apart. Top each onion mound with jalapeño slices (about 1 tablespoon each) and sprinkle with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook without stirring until the onions start to brown, about 30 seconds to 1 minute.

Place one meatball on each onion mound. Cover with the parchment paper square and press down firmly with a sturdy spatula or burger press to flatten into 4-inch-wide patties; remove the parchment paper. Repeat with the remaining meatballs and onion mounds. Season the smashed patties with the remaining 3/4 teaspoon salt and 1/2 teaspoon pepper. Cook undisturbed until the edges of the patties begin to crisp, about 2 to 3 minutes.

Flip the patties so the onion-jalapeño mixture is on top. Add one cheese slice to each of 4 patties. Continue cooking until the patties are well browned and the cheese is melted, about 1 minute and 30 seconds to 2 minutes. Stack each cheese-covered patty on top of a plain patty to create 4 double stacks. Transfer each stack to the bottom half of a toasted bun. Spread about 1 tablespoon of burger sauce on each top bun and place it on the patties. Serve immediately.