Combine 1/4 cup vinegar and sugar in a small bowl, stirring until sugar dissolves.
Add shallots to vinegar mixture; toss. Cover and chill 30 minutes. Toss.
Drain; set aside.
Divide beef into 8 equal portions; gently shape each portion into a 1/4-inch-thick patty, taking care not to pack beef.
Sprinkle both sides of patties evenly with salt and pepper.
Heat a grill pan over medium-high heat. Coat pan with cooking spray.
Add patties to pan; cook 2 minutes on each side or until desired degree of doneness; do not press on patties as they cook.
Combine remaining 1 tablespoon vinegar, oil, mustard, and soy sauce in a large bowl, stirring with a whisk.
Add arugula and chives; toss gently to coat.
Combine mayonnaise and cheese in a small bowl.
Spread bottom half of each bun with about 2 teaspoons mayonnaise mixture. Top each slider with 1 patty, about 1/4 cup arugula mixture, 1 tablespoon pickled shallot, and 1 bun top.