Heat oil in small saucepan over medium-low heat. Add onion and cover. Cook, stirring often until golden, about 4 minutes. Add chile and garlic and cook until fragrant, about 1 minute. Add 1/2 package blueberries and 1/2 package raspberries, vinegar, brown sugar, salt and spices. Bring to a simmer over medium heat. Cook until juices thicken and form large bubbles, about 5 minutes. Let cool. Transfer to blender and purée.

Prepare a medium-hot fire in an outdoor grill. For a gas grill, preheat on High, then adjust to Medium-High (about 450°F). For a charcoal grill, let coals burn for about 15 minutes after white ashes form on surface, and you can hold your hand just above cooking grate for about 3 seconds.

Shape ground sirloin into 6 burgers, each about 3 inches in diameter. Press a deep indentation about 1 inch across into each burger to help reduce shrinkage. Sprinkle on both sides with salt and pepper. Let stand at room temperature while preparing fire, but no longer than 45 minutes.

Lightly oil grill grate. Add burgers and close lid. Turn burgers halfway through cooking time, about 6 minutes total for medium-rare. During last minute or so, add rolls to grill to heat and toast.

Place each burger on a bun with 2 tablespoons Spicy Berry Ketchup and a lettuce leaf. Serve hot with remaining berries.