Prepare your grill for indirect cooking. If you're using charcoal, arrange the hot coals on one side of the grill. With a gas grill, turn one side to high and the other to low.
In a medium bowl, stir together the first eight ingredients. Add the chicken and mix well. Form the mixture into 6 patties. (If the mixture is too moist to shape, you can add up to 2 tablespoons of panko.)
Generously oil the grill grates. Lightly sear the burgers on the warmer side of the grill, about 3 minutes per side. Transfer the burgers to the cooler side of the grill. Cover and cook the burgers until they reach 165° at the center, about 3 minutes more per side.
Top each burger with a cheese slice and grill for a minute or two to melt it. Place each patty on a bun and top with avocado.