Drain 1 jar relish in a sieve; discard liquid. Pulse 2/3 cup drained relish with onion and garlic in a food processor until finely chopped.
Heat 1 tablespoon oil in a skillet over medium-high heat; add relish mixture. Cook, stirring, until a thick paste forms, about 7 minutes. Stir in soy sauce. Transfer to a bowl; let cool 5 minutes. Stir in turkey, Worcestershire, mustard, bread, 1 1/2 teaspoons salt and pepper to taste. Form into 8 patties, each about 4 inches in diameter. Refrigerate, covered, at least 1 hour and up to 8 hours.
Preheat grill to medium. Brush both sides of burgers with oil, then brush grates. Grill 5 minutes. Flip; lightly brush tops with chili sauce and grill until a thermometer inserted into centers registers 165 degrees F, 5 to 6 minutes more.
Combine chili sauce and mayonnaise. Drizzle kale with vinegar and remaining teaspoon oil; season with salt and pepper. Brush cut sides of rolls with butter; grill until lightly toasted. Sandwich burgers in rolls with chili sauce mixture, remaining relish and kale.