1. 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.
2. 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.
3. 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.