Make the sauce: Put the blueberries and 2 tablespoons of the vinegar in a food processor. Pulse the berries just to break them up and get some juices flowing.
In a medium saucepan, combine the pulverized berries with the remaining vinegar, sugar, chile flakes, cinnamon stick, and bay leaf. Bring to a simmer over medium heat, reduce the heat to low, and then cover the pot. Cook for 1 hour, stirring occasionally.
Carefully transfer the hot sauce to a blender. Remove the knob from the top of the blender lid and cover with a dish towel to prevent an explosion. Blend the sauce to get it as smooth as possible, then pass it through a fine mesh strainer, directly back into the pan. Simmer the sauce over medium-low heat, stirring occasionally, until it has reduced by about one-third of its original volume (25–30 minutes). It should coat the back of a spoon and have the viscosity of maple syrup. Let the sauce rest, at least overnight, before using. Blue Q will keep for months, covered, in the refrigerator.
Make the chicken: Light your grill and heat it to roughly 350°. Thirty minutes before you plan to grill, bring the chicken to room temperature and season with 2 teaspoons of kosher salt and the black pepper.
Place chicken, skin side up, on the grill and close the lid. Cook, covered, for 20 minutes. Baste with blueberry sauce every 5 minutes for an additional 20 minutes, keeping the lid closed between basting activity. After a total of 40 minutes, flip chicken over to caramelize the skin, and baste the other side for 10 minutes more. Using a probe thermometer, check the internal temperature of the meaty part of a thigh. When it reaches 165°, remove chicken from the grill.
Off the grill, douse the chicken in 1⁄2 cup of blueberry sauce and let it rest for 10 minutes. Carve into 6 or 8 pieces and toss once more in another 1⁄2 cup sauce. Serve the chicken warm or at room temperature.