Bring vinegar, ½ cup water, 2 tablespoons brown sugar, and 1 tablespoon salt to a boil in a small saucepan over medium-high, stirring often to dissolve sugar. Remove from heat, and stir in carrots and radish. Set aside

Stir together mayonnaise, 1 ½ tablespoons chile-garlic sauce, and ½ teaspoon fish sauce in a small bowl; cover and chill spicy mayonnaise until ready to use.

Stir together pork, ginger, cilantro stems, remaining 1 tablespoon brown sugar, remaining 1 tablespoon chile-garlic sauce, remaining 1 teaspoon fish sauce, and remaining 1 teaspoon salt in a large bowl until just combined. Divide pork mixture into 8 (about 3-ounce) portions, and shape each portion into a loose ball. Place meat balls on a large plate; cover and chill while preheating grill.

Preheat grill to high (450°F to 500°F). Place a griddle or large cast-iron skillet on preheated grill, and let heat for 10 minutes. (Alternatively, preheat a flattop grill to medium-high.) Cut a 3-inch square piece of parchment paper. Working in batches if needed, arrange meat balls on griddle, spacing at least 2 inches apart. Place parchment paper square on top of 1 meat ball, and use a sturdy spatula or burger press to firmly press and flatten, forming a 4-inch-wide patty; remove parchment paper. Repeat with remaining meat balls. Cook, uncovered and undisturbed, until bottom sides of patties are browned, about 2 minutes. Flip patties; cook until browned on both sides, 30 seconds to 1 minute. Remove patties from grill.

Drain pickled carrot mixture; discard brine. Spread spicy mayonnaise on cut sides of buns. Place 2 patties on each bottom bun half, and top evenly with pickled carrot mixture (about ¼ cup each). Top with sliced jalapeño and cilantro leaves to taste. Sandwich with top buns. Serve immediately.