Heat grill to medium-high; lightly oil it. Grill jalapenos until slightly charred and blistered, turning frequently, about 15 minutes.
Using a paper towel, pull off stems and remove skins from chilies. Roughly chop flesh and seeds; set aside.
Meanwhile, grill scallions, turning often, until soft and charred in spots, 3 to 4 minutes; set aside.
Place 2 cups blueberries in a food processor; pulse until coarsely chopped, about five times. Transfer to a medium bowl, and add onion, jalapenos, lime juice, and 1/4 teaspoon salt; stir to combine. Add basil, cilantro, remaining cup blueberries, and oil; stir to combine.
Sprinkle chicken with the remaining 1/4 teaspoon salt and the cayenne. Grill chicken until cooked through, 3 to 4 minutes per side.
Place 4 grilled scallions on each plate, and top with chicken. Serve immediately, topped with a generous spoonful of blueberry salsa. Garnish with basil and cilantro leaves.