Bring vinegar, 1/2 cup water, and sugar to a boil in a small saucepan over high. Combine cucumber and carrots in a heatproof bowl; pour hot vinegar mixture over vegetables. Cover and chill at least 15 minutes or up to 3 days. Stir together mayonnaise, Sriracha, and lime juice; set aside. Preheat grill to medium-high (about 450°F). Combine beef, oil, garlic, ginger, and black pepper in a bowl; gently mix just until ingredients are incorporated. Shape mixture into 4 (1/2-inch-thick) patties. Place patties on a grill grate coated with cooking spray. Grill, covered, until a thermometer inserted in thickest portion registers 140°F, about 3 minutes per side, or to desired degree of doneness. Remove from heat, and keep warm. Spread about 1 tablespoon mayonnaise mixture on each bun half. Place 1 burger patty on each bottom bun half. Drain cucumber mixture; place about 1/4 cup on each burger. Divide cilantro sprigs and chiles among burgers. Cover with top halves of buns.